Propagation of Regularity and Persistence of Decay for Fifth Order Dispersive Models

Abstract

This paper considers the initial value problem for a class of fifth order dispersive models containing the fifth order KdV equation

$$\begin{aligned} \partial _tu - \partial _x^5u -30u^2\partial _xu + 20\partial _xu\partial _x^2u + 10u\partial _x^3u = 0. \end{aligned}$$

The main results show that regularity or polynomial decay of the data on the positive half-line yields regularity in the solution for positive times.

1 Introduction

In this work we study propagation of regularity and persistence of decay results for a class of fifth order dispersive models. For concreteness, the main theorems are stated for initial value problems of the form

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu - \partial _x^5u + c_1u^2\partial _xu + c_2\partial _xu\partial _x^2u + c_3u\partial _x^3u = 0, \qquad x,t\in \mathbb {R}, \\ u(x,0) = u_0(x), \end{array}\right. } \end{aligned}$$

(1.1)

where $c_{j}$ are real constants, $u:\mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$ is an unknown function and $u_0:\mathbb {R}\rightarrow \mathbb {R}$ is a given function. Eq. (1.1) contains the specific equation

$$\begin{aligned} \partial _tu - \partial _x^5u - 30u^2\partial _xu + 20\partial _xu\partial _x^2u + 10u\partial _x^3u = 0 \end{aligned}$$

(1.2)

which is the third equation in the sequence of nonlinear dispersive equations

$$\begin{aligned} \partial _tu + \partial _x^{2j+1}u + Q_j\left( u,\partial _xu,\ldots ,\partial _x^{2j-1}u\right) = 0, \quad j\in \mathbb {Z}^+, \end{aligned}$$

(1.3)

known as the KdV hierarchy. Here the polynomials $Q_j$ are chosen so that Eq. (1.3) has the Lax pair formulation

$$\begin{aligned} \partial _tu = [B_j;L]u \end{aligned}$$

for $L=\frac{d^2}{dx^2}-u(x)$ the Schrödinger operator [16]. The first two equations in the hierarchy are

$$\begin{aligned} \partial _tu - \partial _xu = 0 \end{aligned}$$

(1.4)

and the KdV equation

$$\begin{aligned} \partial _tu + \partial _x^3u + u\partial _xu = 0. \end{aligned}$$

(1.5)

With only slight modifications concerning the hypothesis on the initial data, the techniques in this paper apply to a large class of fifth order equations including the following models arising from mathematical physics:

$$\begin{aligned} \partial _tu + \partial _xu + c_1u\partial _xu + c_2\partial _x^3u + c_3\partial _xu\partial _x^2u + c_4u\partial _x^3u + c_5\partial _x^5u = 0 \end{aligned}$$

(1.6)

modelling the water wave problem for long, small amplitude waves over shallow bottom [22], a model describing short and long wave interaction [1]

$$\begin{aligned} \partial _tu - 2\partial _xu\partial _x^2u - u\partial _x^3u + \partial _x^5u = 0, \end{aligned}$$

(1.7)

and Lisher’s model for motion of a lattice of anharmonic oscillators [18]

$$\begin{aligned} \partial _tu + (u+u^2)\partial _xu +(1+u)\left( \partial _xu\partial _x^2u + u\partial _x^3u\right) + \partial _x^5u = 0. \end{aligned}$$

(1.8)

See also [24] and references therein.

Following Kato’s definition [9], the initial value problem (IVP) (1.1) is said to be locally well-posed in the Banach space X if for every $u_0 \in X$ there exists $T>0$ and a unique solution u(t) satisfying

$$\begin{aligned} u \in C([0,T] ; X) \cap Y_T, \end{aligned}$$

(1.9)

where $Y_T$ is an auxillary function space. Moreover, the solution map $u_0 \mapsto u$ is continuous from X into the class (1.9). If T can be taken arbitrarily large, the IVP (1.1) is said to be globally well-posed. The persistence condition (1.9) states that the solution curve describes a dynamical system.

It is natural to study the IVP (1.1) in the Sobolev spaces

$$\begin{aligned} H^s(\mathbb {R}) = \left( 1-\partial _x^2\right) ^{-s/2}L^2(\mathbb {R}), \quad s\in \mathbb {R}, \end{aligned}$$

having norm

$$\begin{aligned} \Vert f\Vert _{H^s} = \Vert J^sf\Vert _2 \sim \Vert f\Vert _2 + \Vert D^sf\Vert _2. \end{aligned}$$

The homogeneous derivative D and its inhomogeneous counterpart J are defined via the Fourier multipliers

$$\begin{aligned} \widehat{D^sf}(\xi ) = |\xi |^{s}\hat{f}(\xi ) \qquad \text {and}\qquad \widehat{J^sf}(\xi ) = \langle \xi \rangle ^{s}\hat{f}(\xi ), \quad s\in \mathbb {R}, \end{aligned}$$

where $\langle x \rangle = (1+x^2)^{1/2}$. The weighted spaces

$$\begin{aligned} X_{s,m} = H^s(\mathbb {R}) \cap L^2(|x|^m \; dx) \quad s\in \mathbb {R}, m\in \mathbb {Z}^+\cup \{0\} \end{aligned}$$

also appear in our analysis. Additionally, we use the notation $x_+=\max \{0,x\},x_{-}=\min \{0,x\}$ and write $A \lesssim B$ to denote $A \le cB$ when the value of the fixed constant c is immaterial. The floor and ceiling functions are denoted by $\lfloor {x}\rfloor$ and $\lceil {x}\rceil$, respectively.

The persistence property (1.9) doesn’t preclude all smoothing effects. For step-data, Murray [21] proved the existence of solutions to the initial value problem for the KdV Equation (1.5) in the class $C^\infty (\{x,t : x\in \mathbb {R}, \; t>0)\})$ which weakly recover the initial data. Kato [9] described this quasiparabolic smoothing effect as stemming from the unidirectional dispersion inherent in the equation. He obtained a similar result for data having exponential decay on the positive half-line. The Kato estimates occur in the asymmetric spaces

$$\begin{aligned} H^s(\mathbb {R}) \cap L_\beta ^2(\mathbb {R}), \quad s\ge 0,\ \beta >0, \end{aligned}$$

where

$$\begin{aligned} L_\beta ^2(\mathbb {R}) = L^2(e^{\beta x} \; dx), \end{aligned}$$

in which the operator $\partial _t+\partial _x^3$ is formally equivalent to $\partial _t+(\partial _x-\beta )^3$. The use of asymmetric spaces leads to a result which is irreversible in time. Isaza et al. [8] extended the quasiparabolic smoothing effect to a large class of fifth order equations.

Theorem A

(Isaza et al. [8]) Let $u \in C([0,T];H^6(\mathbb {R}))$ be a solution of the IVP associated to the equation

$$\begin{aligned} \partial _tu - \partial _x^5u + Q_0\left( u,\partial _xu,\partial _x^2u\right) \partial _x^3u + Q_1\left( u,\partial _xu,\partial _x^2u\right) = 0 \end{aligned}$$

(1.10)

corresponding to initial data $u_0 \in H^6(\mathbb {R}) \cap L^2(e^{\beta x} \; dx), \beta >0$, with

$$\begin{aligned} Q_0=\sum _{1 \le i+j+k \le N} a_{i,j,k} u^i(\partial _xu)^j\left( \partial _x^2u\right) ^k, \quad N\in \mathbb {Z}^+, N \ge 1,\ a_{i,j,k}\in \mathbb {R} \end{aligned}$$

(1.11)

and

$$\begin{aligned} Q_1=\sum _{2 \le i+j+k \le M} b_{i,j,k} u^i(\partial _xu)^j\left( \partial _x^2u\right) ^k \quad M\in \mathbb {Z}^+, M \ge 2,\,b_{i,j,k}\in \mathbb {R}. \end{aligned}$$

(1.12)

Then

$$\begin{aligned} e^{\beta x}u \in C([0,T];L^2(\mathbb {R})) \cap C((0,T);H^\infty (\mathbb {R})), \end{aligned}$$

and

$$\begin{aligned} \Vert e^{\beta x}u(t)\Vert _2 \le c\Vert e^{\beta x}u_0\Vert _2, \quad t\in [0,T]. \end{aligned}$$

Kato [9] demonstrated the existence of weak global solutions u to the KdV Equation (1.5) corresponding to initial data in $L^2(\mathbb {R})$. A key step in his proof is the a priori estimate of $\Vert u\Vert _{H^1(-R,R)}$ in terms of $\Vert u_0\Vert _2$. In addition, his approach shows the following local smoothing effect.

Theorem B

(Kato [9]) Let $s>3/2$ and $0<T<\infty$. If $u \in C([0,T]; H^s(\mathbb {R}))$ is the solution to (1.5), then

$$\begin{aligned} u \in L^2([0,T];H^{s+1}(-R,R)) \quad \text {for any}\,\,0<R<\infty , \end{aligned}$$

with the associated norm depending only on $\Vert u_0\Vert _{H^s}$, R and T.

Roughly, the proof follows by observing that a smooth solution u to the IVP associated to the KdV Equation (1.5) satisfies the identity

$$\begin{aligned}&\frac{d}{dt} \int \left( \partial _x^ku\right) ^2\psi \; dx + 3 \int \left( \partial _x^{k+1}u\right) ^2\psi ' \; dx\nonumber \\&\quad = \int \left( \partial _x^ku\right) ^2\psi ''' \; dx + \int \partial _x(\psi u)\left( \partial _x^ku\right) ^2 \; dx + \int \partial _x^ku \left[ \partial _x^k;u\right] \partial _xu\psi \; dx. \end{aligned}$$

(1.13)

for $k\in \mathbb {Z}^+$. Selecting $\psi =\psi (x)$ to be a sufficiently smooth, nonnegative, nondecreasing cutoff function, integration of the above identity in time yields local estimates of $\partial _x^{k+1}u$ as each term on right-hand side can be controlled by $\Vert u\Vert _{L_T^\infty H^k}$.

Isaza et al. applied Kato’s argument to study the propagation of regularity and persistence of decay of solutions to the k-generalized KdV and Benjamin–Ono equations in Refs. [7] and [6], respectively. Also working in asymmetric spaces, they observed that for a solution u to the KdV equation corresponding to data $u_0 \in H^{s}(\mathbb {R})$ with $s>3/4$, if $\Vert x^{n/2}u_0\Vert _{L^2(0,\infty )}$ for some $n\in \mathbb {Z}^+$, then for every $x_0\in \mathbb {R}$, $u(\cdot ,t) \in H^n(x_0,\infty )$ for positive times. More succinctly, one-sided decay on the initial data yields regularity in the solution. In this paper we extend their work to fifth order dispersive models. Before stating our results we review the local well-posedness theory for (1.1) and related models.

Utilizing the Lax pair formulation, initial value problems associated to equations in the KdV hierarchy (1.3) can be solved in a space of rapidly decaying functions using the inverse scattering method [4]. This method does not apply to dispersive equations of a more general form.

While studying the models (1.1), (1.6), (1.7) and (1.8), Ponce [24] remarked that the use of dispersive estimates appears essential to attain local well-posedness in Sobolev spaces. Using the energy method, sharp linear estimates and parabolic regularization, in Ref. [24] Ponce proved local well-posedness for the initial value problems associated to these equations in $H^s(\mathbb {R})$, $s\ge 4$.

Kenig et al. investigated the class

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu + \partial _x^{2j+1}u + P\left( u,\partial _xu,\ldots ,\partial _x^{2j}u\right) = 0, \qquad x,t\in \mathbb {R},\\ u(x,0) = u_0(x), \end{array}\right. } \end{aligned}$$

(1.14)

with $j\in \mathbb {Z}^+$ and $P : \mathbb {R}^{2j+1}\rightarrow \mathbb {R}$ (or $\mathbb {C}^{2j+1}\rightarrow \mathbb {C}$) a polynomial having no constant or linear terms. Using the contraction principle, they established in Refs. [12] and [11] that for a given equation in the class (1.14) there exists a positive real number $s_0$ and nonnegative integer $m_0$ depending only on the form of the polynomial P such that the corresponding IVP is locally well-posed in the weighted space $X_{s,m}$ for all $m\in \mathbb {Z}^+$, $m \ge m_0$ and $s \ge \max \{s_0,jm\}$. Thus equations of the form (1.14) preserve the Schwarz class. The use of weighted spaces stems from the observation that $[L;\varGamma ]=0$ for the vector fields

$$\begin{aligned} L=\partial _t + \partial _x^{2j+1} \quad \text {and}\quad \varGamma = x - (2j+1)t \partial _x^{2j}. \end{aligned}$$

Given that each term of P has “enough” factors, it may be that the corresponding IVP is globally well-posed, that no weight is necessary or both. For further comments, see [17].

Following [19] and [20], Pilod [23] showed that certain initial value problems in the class (1.14) are in some sense ill-posed. In particular, if P contains the term $u\partial _x^ku$ for $k>j$, then the solution map $H^s(\mathbb {R}) \ni u_0 \mapsto u \in C([0,T];H^s(\mathbb {R}))$ is not $C^2$ at the origin for any $s\in \mathbb {R}$. For equations of the form (1.1), Kwon demonstrated that the solution map is not even uniformly continuous by using the arguments of [13] and [14]. All of these facts result from uncontrollable interactions when both high and low frequencies are present in the initial data. Thus, in contrast to the KdV (1.5), equations of the form (1.1) cannot be solved using the contraction principle in $H^s(\mathbb {R})$.

Differences between (1.1) and (1.5) also arise when applying the energy estimate method. Note that after integrating by parts, smooth solutions u to (1.1) satisfy

$$\begin{aligned}&\frac{d}{dt} \int \left( \partial _x^ku\right) ^2\psi (x) \; dx + 2 \int \left( \partial _x^{k+2}u\right) ^2\psi ' \; dx \nonumber \\&\quad \lesssim \left\| \partial _x^3u\right\| _\infty \int \left( \partial _x^ku\right) ^2\psi (x) \; dx + \left| \int \partial _xu\left( \partial _x^{k+1}u\right) ^2\psi \; dx\right| + \cdots \end{aligned}$$

(1.15)

for $k\in \mathbb {Z}^+$. After integrating in time, the right-hand side cannot be estimated in terms of $\Vert u\Vert _{L_T^\infty H^k}$. Kwon [15] introduced a corrected energy and refined Strichartz estimate to overcome this loss of derivatives and obtained the following result.

Theorem C

(Kwon [15]) Let $s>5/2$. For any $u_0 \in H^s(\mathbb {R})$ there exists a time $T \gtrsim \Vert u_0\Vert _{H^s}^{-10/3}$ and a unique real-valued solution u for the IVP (1.1) satisfying

$$\begin{aligned} u \in C([0,T];H^s(\mathbb {R})) \qquad \text {and}\qquad \partial _x^3u \in L^1([0,T];L^\infty (\mathbb {R})). \end{aligned}$$

(1.16)

Remark 1

A loss of derivatives can occur for equations for which LWP can be obtained in $H^s(\mathbb {R})$ using the contraction principle (see Sect. 7).

Using an auxillary Bourgain space introduced in Refs. [2, 3], the local well-posedness of the IVP (1.1) in the energy space $H^2(\mathbb {R})$ was established simultaneously by Kenig and Pilod [10] and Guoet al. [5]. Thus global well-posedness follows in the Hamiltonian case, i.e., when $c_{2}=2c_{3}$.

Our main contribution is the incorporation of Kwon’s corrected energy and refined Strichartz estimate into the iterative argument used in Refs. [7] and [6]. We first describe the propagation of one-sided regularity exhibited by solutions to the IVP (1.1) provided by Theorem C.

Theorem 1

Let $s>5/2$. Suppose $u_0 \in H^{s}(\mathbb {R})$ and for some $l \in \mathbb {Z}^+, x_0 \in \mathbb {R}$

$$\begin{aligned} \left\| \partial _x^lu_0\right\| _{L^2(x_0,\infty )}^2 = \int _{x_0}^\infty \left( \partial _x^lu_0\right) ^2(x) \; dx < \infty . \end{aligned}$$

(1.17)

Then the solution u of IVP (1.1) provided by Theorem C satisfies

$$\begin{aligned} \sup _{0 \le t \le T} \int _{x_0 + \varepsilon - \nu t}^\infty \left( \partial _x^m u\right) ^2(x,t) \; dx \le c \end{aligned}$$

(1.18)

for any $\nu \ge 0, \varepsilon >0$ and each $m=0,1,\ldots ,l$ with

$$\begin{aligned} c = c\left( l; \nu ; \varepsilon ; T; \Vert u_0\Vert _{H^s}; \left\| \partial _x^lu_0\right\| _{L^2(x_0,\infty )}\right) , \end{aligned}$$

(1.19)

where T is given in Theorem C. In particular, for all $t\in (0,T]$, the restriction of $u(\cdot ,t)$ to any interval $(x_1,\infty )$ belongs to $H^l(x_1,\infty )$.

Moreover, for any $\nu \ge 0, \varepsilon >0$ and $R>\varepsilon$

$$\begin{aligned} \int _0^T \int _{x_0 + \varepsilon - \nu t}^{x_0 + R - \nu t} \left( \partial _x^{l+2}u\right) ^2(x,t) \; dxdt \le \tilde{c} \end{aligned}$$

(1.20)

with

$$\begin{aligned} \tilde{c} = \tilde{c}\left( l; \nu ; \varepsilon ; R; T; \Vert u_0\Vert _{H^s}; \left\| \partial _x^lu_0\right\| _{L^2(x_0,\infty )}\right) . \end{aligned}$$

(1.21)

Remark 2

Observe that (1.20) is a generalization of Kato’s local smoothing effect since we do not require $u_0 \in H^l(\mathbb {R})$.

Remark 3

The constants appearing in Theorem 1 have the form of a polynomial in $\nu$. For $l\ge 6$, the degree of this dependence is $d=8(l-5)$.

For fixed $l\in \mathbb {Z}^+$, Theorem 1 is the base case for the situation where the derivatives of the initial data possess polynomial decay when restricted to the positive half-line. Our second result states that this decay persists.

Theorem 2

Let $s>5/2$ and let $n,l\in \mathbb {Z}^+$. Suppose $u_0 \in H^{s}(\mathbb {R})$ and for each $m=0,1,\ldots ,l$

$$\begin{aligned} \left\| x^{n/2}\partial _x^mu_0\right\| _{L^2(0,\infty )}^2 = \int _0^\infty x^n\left( \partial _x^mu_0\right) ^2(x) \; dx < \infty . \end{aligned}$$

(1.22)

Then the solution u of IVP (1.1) provided by Theorem C satisfies

$$\begin{aligned} \sup _{0 \le t \le T} \int _\varepsilon ^\infty x^n\left( \partial _x^mu\right) ^2(x,t) \; dx \le c \end{aligned}$$

(1.23)

for any $\varepsilon >0$ and each $m=0,1,\ldots ,l$ with

$$\begin{aligned} c = c\left( n; l; \varepsilon ; T; \Vert u_0\Vert _{H^s}; \left\| x^{n/2}\partial _x^ku_0\right\| _{L^2(0,\infty )}\right) \end{aligned}$$

(1.24)

for $k=0,1,\ldots ,m$, where T is given in Theorem C. By local well-posedness, we may take $\varepsilon =0$ for $m \le s$.

Moreover, for any $\varepsilon >0$

$$\begin{aligned} \int _0^T \int _0^\infty x^{n-1}\left( \partial _x^{l+2}u\right) ^2(x,t) \; dxdt \le \tilde{c} \end{aligned}$$

(1.25)

with $\tilde{c}$ as in (1.24).

The hypothesis of Theorem 2 may seem unneccessarily strong, but a bootstrapping argument yields regularity of the solution for positive times by imposing decay on only the initial data and not its derivatives. Thus the next theorem can be seen as a weakening of the hypothesis of Theorem A in as much as exponential decay implies polynomial decay.

Theorem 3

Let $s>5/2$. Suppose $u_0 \in H^s(\mathbb {R})$ and for some $n \in \mathbb {Z}^+$

$$\begin{aligned} \Vert x^{n/2}u_0\Vert _{L^2(0,\infty )}^2 = \int _0^\infty x^nu_0^2(x) \; dx < \infty . \end{aligned}$$

(1.26)

Then for every $\delta >0$ and any pair $m,k \in \mathbb {Z}^+\cup \{0\}$ satisfying

$$\begin{aligned} n=k+\lfloor m/2 \rfloor \end{aligned}$$

(1.27)

the solution u of IVP (1.1) provided by Theorem C satisfies, for $k>0$

$$\begin{aligned} \sup _{\delta \le t \le T} \int _{\varepsilon -\nu t}^\infty \left( \partial _x^mu\right) ^2(x,t) \langle x_+\rangle ^k \; dx + \int _\delta ^T \int _{\varepsilon -\nu t}^\infty \left( \partial _x^{m+2}u\right) ^2(x,t)\langle x_+\rangle ^{k-1} \; dxdt \le c\nonumber \\ \end{aligned}$$

(1.28)

for every $\nu \ge 0, \varepsilon >0$, with

$$\begin{aligned} c = c\left( n; \delta ; \nu ; \varepsilon ; T; \Vert u_0\Vert _{H^s}; \Vert x^{n/2}u_0\Vert _{L^2(0,\infty )}\right) , \end{aligned}$$

(1.29)

where T is given in Theorem C. For $k=0$ and any $R>\varepsilon$,

$$\begin{aligned} \sup _{\delta \le t \le T} \int _{\varepsilon -\nu t}^\infty \left( \partial _x^{2n}u\right) ^2(x,t) \; dx + \int _\delta ^T \int _{\varepsilon -\nu t}^{R-\nu t} \left( \partial _x^{2n+2}u\right) ^2(x,t) \; dxdt \le \tilde{c} \end{aligned}$$

(1.30)

with $\tilde{c}$ additionally depending on R.

The time reversible nature of Eq. (1.1) yields a number of consequences. Combining with the contrapositive of Theorems 1 and 3, we have the following.

Corollary 1

Assume that $s>5/2$. Let $u \in C([-T,T];H^{s}(\mathbb {R}))$ be a solution of (1.1) provided by Theorem C such that

$$\begin{aligned} \partial _x^mu(\cdot ,\hat{t}) \notin L^2(a,\infty ) \quad \text {for some}\,\hat{t}\in [-T,T]\,\hbox {and}\,a\in \mathbb {R}. \end{aligned}$$

Then for any $t\in [-T,\hat{t})$ and any $\beta \in \mathbb {R}$

$$\begin{aligned} \partial _x^mu(\cdot ,t) \notin L^2(\beta ,\infty ) \quad \text {and}\quad x^{\lceil {m/2}\rceil /2}u(\cdot ,t) \notin L^2(0,\infty ). \end{aligned}$$

Suppose now that the initial data has regularity to the right but also contains a singularity, for instance $u_0 \in H^{s}(\mathbb {R})$, $u_0 \notin H^l(\mathbb {R})$ and

$$\begin{aligned} \partial _x^lu_0 \in L^2(b,\infty ) \quad \text {for some}\,l\in \mathbb {Z}^+,l>2. \end{aligned}$$

The persistence property (1.9) prohibits the solution from lying in $H^l(\mathbb {R})$. However, as a consequence of Remark 3, we deduce that for positive times $\partial _x^lu(\cdot ,t)$ has only polynomial growth to the left and thus lies in $L^2_{\text {loc}}(\mathbb {R})$. That is, any singularities in $\partial _x^lu(\cdot ,t)$ vanish for positive times. This is made precise by the next corollary to Theorem 1.

Corollary 2

Assume that $s>5/2$. Let $u \in C([-T,T];H^{s}(\mathbb {R}))$ be a solution of (1.1) provided by Theorem C. Suppose there exists $l,m\in \mathbb {Z}^+$ with $m \le l$ such that for some $a,b\in \mathbb {R}$ with $a<b$

$$\begin{aligned} \int _b^\infty \left( \partial _x^lu_0\right) ^2(x) \; dx < \infty \quad \text {but}\quad \partial _x^mu_0 \notin L^2(a,\infty ). \end{aligned}$$

(1.31)

• (i) For any $t\in (0,T]$ and any $\varepsilon >0$

  $$\begin{aligned} \int _{-\infty }^\infty \frac{1}{\langle x_{-} \rangle ^{8(l-5)+\varepsilon }} \left( \partial _x^lu\right) ^2(x,t) \; dx \le c, \quad l\ge 6 \end{aligned}$$

  (1.32)

  with c depending on t and $\varepsilon$.

• (ii) For any $t\in [-T,0)$ and any $\alpha \in \mathbb {R}$

  $$\begin{aligned} \int _\alpha ^\infty \left( \partial _x^mu\right) ^2(x,t) \; dx = \infty . \end{aligned}$$

Remark 4

The conclusion (1.32) holds for $l=3,4,5$ with the appropriate modification to the weight.

As a consequence of Corollary 2 we see that, in general, regularity to the left does not propagate forward in time. Suppose in addition to (1.31) that

$$\begin{aligned} \int _{-\infty }^a\left( \partial _x^lu_0\right) ^2(x) \; dx < \infty . \end{aligned}$$

If this regularity persisted we could conclude from (1.32) that $u(\cdot ,t) \in H^l(\mathbb {R})$ for positive times, contradicting the persistence property (1.9).

Beginning with Theorem 3 yields a similar corollary.

Corollary 3

Assume that $s>5/2$. Let $u \in C([-T,T];H^{s}(\mathbb {R}))$ be a solution of (1.1) provided by Theorem C. If for $m,n\in \mathbb {Z}^+$, $m<n$,

$$\begin{aligned} x_+^{\lceil {n/2}\rceil /2}u_0 \in L^2(0,\infty ) \quad \text {and}\quad \partial _x^mu_0 \notin L^2(\beta ,\infty ) \quad \text {for some}\,\beta \in \mathbb {R}, \end{aligned}$$

then for any $t\in (0,T]$

$$\begin{aligned} x_+^{\lceil {n/2}\rceil /2}u(\cdot ,t) \in L^2(0,\infty ) \quad \text {and}\quad \partial _x^nu(\cdot ,t) \in L^2(\alpha ,\infty ) \quad \text {for any}\,\alpha \in \mathbb {R}, \end{aligned}$$

and for any $t\in [-T,0)$

$$\begin{aligned} x_+^{\lceil {m/2}\rceil /2}u(\cdot ,t) \notin L^2(0,\infty ) \quad \text {and}\quad \partial _x^mu(\cdot ,t) \notin L^2(\alpha ,\infty ) \quad \text {for any}\,\alpha \in \mathbb {R}. \end{aligned}$$

Our proof technique does not rely on the particular values of the coefficients in (1.1), hence Theorems 1, 2 and 3 can be applied backwards in time. For instance, if u(x, t) is a solution of (1.1) with regularity to the right which propagates leftward, then $u(-x,-t)$ has regularity to the left which propagates rightward. Therefore we can consider the situation when $u(\cdot ,t_0)$ has decay or regularity to the right and $u(\cdot ,t_1)$ has decay or regularity to the left, where $t_0<t_1$.

Corollary 4

Assume that $s>5/2$. Let $u \in C([-T,T];H^{s}(\mathbb {R}))$ be a solution of (1.1) provided by Theorem C. If there exist $n_j\in \mathbb {Z}^+\cup \{0\}$, $j=1,2,3,4$, $t_0,t_1\in [-T,T]$ with $t_0<t_1$ and $a,b\in \mathbb {R}$ such that

$$\begin{aligned} \int _0^\infty |x|^{n_1}|u(x,t_0)|^2 \; dx < \infty \quad \text {and}\quad \int _a^\infty \left| \partial _x^{n_2}u(x,t_0)\right| ^2 \; dx < \infty \end{aligned}$$

and

$$\begin{aligned} \int _{-\infty }^0 |x|^{n_3}|u(x,t_1)|^2 \; dx < \infty \quad \text {and}\quad \int _{-\infty }^b \left| \partial _x^{n_4}u(x,t_1)\right| ^2 \; dx < \infty \end{aligned}$$

then

$$\begin{aligned} u \in C([-T,T];H^s(\mathbb {R}) \cap L^2(|x|^r \; dx)) \end{aligned}$$

where

$$\begin{aligned} s = \min \left\{ \max \{2n_1,n_2\},\max \{2n_3,n_4\}\right\} \quad \text {and}\quad r=\min \{n_1,n_3\}. \end{aligned}$$

In Sect. 2 we construct cutoff functions which are needed to prove Theorems 1, 2 and 3. Theorems 1 and 2 are proved in Sects. 3 and 4, respectively. In Sect. 5 we prove Theorem 3. The proof of Corollary 2 is found in Sect. 6. We conclude in Sect. 7 with an extension to a more general class of fifth order models.

2 Construction of Cutoff Function

In this section we construct cutoff functions which are needed to prove Theorems 1, 2 and 3. Define the polynomial

$$\begin{aligned} \rho (x) = 2772 \int _0^x y^5(1-y)^5 \, dy \end{aligned}$$

which satisfies

$$\begin{aligned} \rho (0)= & {} 0, \qquad \rho (1)=1, \\ \rho '(0)= & {} \rho ''(0) = \cdots = \rho ^{(5)}(0) = 0, \\ \rho '(1)= & {} \rho ''(1) = \cdots = \rho ^{(5)}(1) = 0 \end{aligned}$$

with $0<\rho ,\rho '$ for $0<x<1$. Much of the complexity of our construction airses when handling the ratio which appears in (3.2), see Sect. 3 below. Thus we note that the expression

$$\begin{aligned} \frac{(\rho '''(x))^2}{\rho '(x)} = -277200 x (x-1) \left( 2-9 x+9 x^2\right) ^2 \end{aligned}$$

(2.1)

is continuous for $x\in [0,1]$ and vanishes at the endpoints. For $\varepsilon ,b>0$, define $\chi \in C^5(\mathbb {R})$ by

$$\begin{aligned} \chi (x;\varepsilon ,b) = {\left\{ \begin{array}{ll} 0 &{} x \le \varepsilon , \\ \rho ((x-\varepsilon )/b) &{} \varepsilon < x < b+\varepsilon , \\ 1 &{} b + \varepsilon \le x. \end{array}\right. } \end{aligned}$$

By construction $\chi$ is positive for $x\in (\varepsilon ,\infty )$ and all derivatives are supported in $[\varepsilon ,b+\varepsilon ]$. A scaling argument and (2.1) provides

$$\begin{aligned} \sup _{x\in [\varepsilon ,b+\varepsilon ]} \left| \frac{(\chi '''(x;\varepsilon ,b))^2}{\chi '(x;\varepsilon ,b)} \right| \le c(b) \end{aligned}$$

(2.2)

and for $j=1,2,3,4,5$

$$\begin{aligned} |\chi ^{(j)}(x;\varepsilon ,b)| \le c(j;b). \end{aligned}$$

(2.3)

A computation produces

$$\begin{aligned} \frac{(\chi '''(x;\varepsilon ,b))^2}{\chi '(x;\varepsilon ,b)}\cdot \frac{1}{\chi '(x;\varepsilon /3,b+\varepsilon )} = q_0(x)\frac{(x-\varepsilon )(b+\varepsilon -x)}{(3 x-\varepsilon )^5 (3 b-3 x+4 \varepsilon )^5} \end{aligned}$$

and for $j=1,2,3,4,5$

$$\begin{aligned} \frac{\chi ^{(j)}(x;\varepsilon ,b)}{\chi '(x;\varepsilon /3,b+\varepsilon )} = q_j(x)\frac{(x-\varepsilon )(b+\varepsilon -x)}{(3 x-\varepsilon )^5 (3 b-3 x+4 \varepsilon )^5} \end{aligned}$$

where $q_0,\ldots ,q_5$ are polynomials. In each of the previous two cases, the right-hand side is continuous on the interval $x\in [\varepsilon ,b+\varepsilon ]$, hence bounded. These computations lead to the following estimates, which will be used in a later inductive argument:

$$\begin{aligned} \sup _{x\in [\varepsilon ,b+\varepsilon ]} \left| \frac{\left( \chi '''(x;\varepsilon ,b)\right) ^2}{\chi '(x;\varepsilon ,b)}\right| \le c(\varepsilon ;b)\chi '(x;\varepsilon /3,b+\varepsilon ) \end{aligned}$$

(2.4)

and for $j=1,2,3,4,5$

$$\begin{aligned} \sup _{x\in [\varepsilon ,b+\varepsilon ]} \left| \chi ^{(j)}(x;\varepsilon ,b)\right| \le c(j;\varepsilon ;b)\chi '(x;\varepsilon /3,b+\varepsilon ). \end{aligned}$$

(2.5)

Additionally, we define $\chi _n \in C^5(\mathbb {R})$ via the formula

$$\begin{aligned} \chi _n(x;\varepsilon ,b) = x^n\chi (x;\varepsilon ,b). \end{aligned}$$

It is helpful to make the auxillary definition

$$\begin{aligned} p(y) = 462 - 1980y + 3465y^2 - 3080y^3 + 1386y^4 - 252y^5, \end{aligned}$$

whose only real root occurs at $y\approx 1.29727$. Note that for $n\in \mathbb {Z}^+$

$$\begin{aligned} \chi _n'(x;\varepsilon ,b) = nx^{n-1}\chi (x;\varepsilon ,b) + x^n\chi '(x;\varepsilon ,b) \end{aligned}$$

(2.6)

which is positive for $\varepsilon < x \le b+\varepsilon$. Hence the expression

$$\begin{aligned} \frac{\left( \chi _n'''(x;\varepsilon ,b)\right) ^2}{\chi _n'(x;\varepsilon ,b)} \end{aligned}$$

is continuous in this interval. To prove that it is bounded in $[\varepsilon ,b+\varepsilon ]$, we must only analyze the limit $x \rightarrow \varepsilon ^+$. First observe

$$\begin{aligned} \chi _n'(x;\varepsilon ,b) = \left( \frac{x-\varepsilon }{b}\right) ^5 \left( \frac{n}{b}x^{n-1}(x-\varepsilon )p\left( \frac{x-\varepsilon }{b}\right) + \frac{2772}{b}x^n \left( 1-\frac{x-\varepsilon }{b}\right) ^5 \right) \end{aligned}$$

so that

$$\begin{aligned} \lim _{x\rightarrow \varepsilon ^+} \frac{\left( \chi _n'''(x;\varepsilon ,b)\right) ^2}{\chi _n'(x;\varepsilon ,b)} = \left( \frac{b^6}{2772\varepsilon ^n}\right) \lim _{x\rightarrow \varepsilon ^+} \frac{\left( \chi _n'''(x;\varepsilon ,b)\right) ^2}{(x-\varepsilon )^5}. \end{aligned}$$

Each term of $\chi _n'''$ has a factor of $(x-\varepsilon )^3$ implying the above limit vanishes. Hence

$$\begin{aligned} \sup _{x\in [\varepsilon ,b+\varepsilon ]} \left| \frac{\left( \chi _n'''(x;\varepsilon ,b)\right) ^2}{\chi _n'(x;\varepsilon ,b)} \right| \le c(n;b) \end{aligned}$$

(2.7)

and so

$$\begin{aligned} \left| \frac{\left( \chi _n'''(x;\varepsilon ,b)\right) ^2}{\chi _n'(x;\varepsilon ,b)} \right| \le c(n;b)(1+\chi _n(x;\varepsilon ,b)). \end{aligned}$$

(2.8)

Each term of (2.6) is nonnegative and $\chi '$ is supported in $[\varepsilon ,b+\varepsilon ]$, hence

$$\begin{aligned} \chi _n'(x;\varepsilon ,b) \le c(n;b)(1+\chi _n(x;\varepsilon ,b)). \end{aligned}$$

Using the Leibniz rule, it similarly follows for $j=1,2,3,4,5$ that

$$\begin{aligned} \left| \chi _n^{(j)}(x;\varepsilon ,b)\right| \le c(n;j;b)(1+\chi _n(x;\varepsilon ,b)). \end{aligned}$$

(2.9)

Assuming $n\ge 3$, notice that (2.7) and

$$\begin{aligned} \frac{\left( \chi _n'''(x;\varepsilon ,b)\right) ^2}{\chi _n'(x;\varepsilon ,b)} = (n-1)(n-2)x^{n-5} \qquad (b+\varepsilon \le x) \end{aligned}$$

imply

$$\begin{aligned} \left| \frac{\left( \chi _n'''(x;\varepsilon ,b)\right) ^2}{\chi _n'(x;\varepsilon ,b)} \right| \le c(n;\varepsilon ;b)\chi _{n-1}(x;\varepsilon /3,b+\varepsilon ). \end{aligned}$$

(2.10)

A similar argument holds for $n=1,2$. Next we prove for $j=1,2,3,4,5$

$$\begin{aligned} \left| \chi _n^{(j)}(x;\varepsilon ,b)\right| \le c(n;j;\varepsilon ;b) \chi _{n-1}(x;\varepsilon /3,b+\varepsilon ). \end{aligned}$$

(2.11)

This follows by definition when $b+\varepsilon \le x$; thus it suffices to prove

$$\begin{aligned} \sup _{x\in [\varepsilon ,b+\varepsilon ]} \left| \frac{\chi _n^{(j)}(x;\varepsilon ,b)}{\chi _{n-1}(x;\varepsilon /3,b+\varepsilon )}\right| \le c(n,j,\varepsilon ,b). \end{aligned}$$

We demonstrate the details for $j=1$, the remaining cases being similar. In this case

$$\begin{aligned} \frac{\chi _n^{(j)}(x;\varepsilon ,b)}{\chi _{n-1}(x;\varepsilon /3,b+\varepsilon )} = \frac{n\chi (x;\varepsilon ,b)}{\chi (x;\varepsilon /3,b+\varepsilon )} + \frac{x\chi '(x;\varepsilon ,b)}{\chi (x;\varepsilon /3,b+\varepsilon )}. \end{aligned}$$

Assuming $\varepsilon \le x \le b+\varepsilon$,

$$\begin{aligned} \frac{n\chi (x;\varepsilon ,b)}{\chi (x;\varepsilon /3,b+\varepsilon )} = n\left( \frac{b+\varepsilon }{b}\right) ^6 \frac{(x-\varepsilon )^6p\left( \frac{x-\varepsilon }{b}\right) }{(x-\frac{\varepsilon }{3})^6p\left( \frac{x-\frac{\varepsilon }{3} }{b+\varepsilon }\right) }. \end{aligned}$$

Note that $\frac{x-\frac{\varepsilon }{3}}{b+\varepsilon }<1$ so that p does not vanish in $[\varepsilon ,b+\varepsilon ]$. Hence this above expression is continuous and bounded on this interval. Similarly for the second term

$$\begin{aligned} \frac{x\chi '(x;\varepsilon ,b)}{\chi (x;\varepsilon /3,b+\varepsilon )} = \frac{2772(b+\varepsilon )^6(x-\varepsilon )^5(b-x+\varepsilon )^5x}{b^{11}(x-\frac{\varepsilon }{3})p\left( \frac{x-\frac{\varepsilon }{3} }{b+\varepsilon }\right) }. \end{aligned}$$

This proves (2.11) in the case $j=1$.

3 Proof of Theorem 1

In this section, we prove Theorem 1. We show several lemmas which are needed to prove Theorems 1, 2 and 3. The first lemma is an analogue of (1.13) to implement Kato’s energy estimate argument which is proved by Isaza et al. [8].

Lemma 1

Let $u \in C^{\infty }([0,T];H^\infty (\mathbb {R}))$ be a solution to IVP

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu - \partial _x^5u = F \qquad x,t\in \mathbb {R} \\ u(x,0) = u_0(x) \end{array}\right. } \end{aligned}$$

(3.1)

and let $\psi \in C^5(\mathbb {R}^2)$ satisfy $\partial _x\psi \ge 0$. Then we have

$$\begin{aligned}&\frac{d}{dt} \int u^2\psi \; dx + \int \left( \partial _x^2u\right) ^2\partial _x\psi \; dx\nonumber \\&\quad \le \int u^2\left\{ \partial _t\psi + \frac{3}{2}\partial _x^5\psi + \frac{25}{16}\frac{\left( \partial _x^3\psi \right) ^2}{\partial _x\psi }\right\} \; dx + 2 \int u F \psi \; dx. \end{aligned}$$

(3.2)

By interpolation we have the following lemma, which is required to apply the inductive hypothesis.

Lemma 2

Suppose $u_0\in L^2(\mathbb {R})$ and for some $l\in \mathbb {Z}^+$, $l\ge 2$, $x_0\in \mathbb {R}$

$$\begin{aligned} \left\| \partial _x^lu_0\right\| _{L^2(x_0,\infty )}^2 = \int _{x_0}^\infty \left| \partial _x^lu_0\right| ^2 \; dx < \infty . \end{aligned}$$

(3.3)

For any $k=1,2,\ldots ,l-1$ and $\delta >0$

$$\begin{aligned} \left\| \partial _x^ku_0\right\| _{L^2(x_0+\delta ,\infty )}^2 = \int _{x_0+\delta }^\infty \left| \partial _x^ku_0\right| ^2 \; dx < \infty . \end{aligned}$$

(3.4)

We reproduce for convenience a lemma of Isaza et al. [7].

Lemma 3

Let $j_1,j_2,j_3\in \mathbb {Z}^+$ and $\varepsilon ,b>0$. Suppose $\psi (x;\varepsilon ,b)$ has support in $[\varepsilon ,\infty )$, $\psi \ge 0$ and $\psi (x;\varepsilon ,b)\ge 1$ whenever $x\ge b+\varepsilon$. Then

$$\begin{aligned}&\int \left| \partial _x^{j_1}u\partial _x^{j_2}u\partial _x^{j_3}u\right| \psi (x) \; dx \nonumber \\&\quad \lesssim \left\{ \int \left( \partial _x^{1+j_1}u\right) ^2\psi (x) \; dx + \int \left( \partial _x^{j_1}u\right) ^2\psi (x) \; dx + \int \left( \partial _x^{j_1}u\right) ^2|\psi '(x)| \; dx\right\} \nonumber \\&\qquad \, \times \int \left( \partial _x^{j_2}u\right) ^2\psi (x;\varepsilon /5,4\varepsilon /5) \; dx + \int \left( \partial _x^{j_3}u\right) ^2\psi (x) \; dx. \end{aligned}$$

(3.5)

In particular, we may choose $\psi =\chi , \chi ', \chi _n$ or $\chi _n'$.

Proof

Using Cauchy–Schwarz and Young’s inequality, followed by the Sobolev embedding, we have

$$\begin{aligned}&\int \left| \partial _x^{j_1}u\partial _x^{j_2}u\partial _x^{j_3}u\right| \psi \; dx\\&\quad \le \frac{1}{2}\int \left( \partial _x^{j_1}u\right) ^2\left( \partial _x^{j_2}u\right) ^2\psi \; dx + \frac{1}{2}\int \left( \partial _x^{j_3}u\right) ^2\psi \; dx \\&\quad \le \frac{1}{2}\left\| \left( \partial _x^{j_1}u\right) ^2\psi \right\| _{L_x^\infty } \int _\varepsilon ^\infty \left( \partial _x^{j_2}u\right) ^2 \; dx + \frac{1}{2}\int \left( \partial _x^{j_3}u\right) ^2\psi \; dx \\&\quad \le \frac{1}{2}\left\| \partial _x\left( \left( \partial _x^{j_1}u\right) ^2\psi \right) \right\| _{L_x^1} \int \left( \partial _x^{j_2}u\right) ^2\psi (x;\varepsilon /5,4\varepsilon /5) \; dx + \frac{1}{2}\int \left( \partial _x^{j_3}u\right) ^2\psi \; dx \end{aligned}$$

since $\psi (x;\varepsilon ,b)$ is nonnegative, supported on $[\varepsilon ,\infty )$ and $\psi (x;\varepsilon ,b)\ge 1$ when $x \ge b+\varepsilon$. Furthermore, Young’s inequality yields

$$\begin{aligned} \left\| \partial _x\left( \left( \partial _x^{j_1}u\right) ^2\psi \right) \right\| _{L_x^1}\le & {} 2\int \left| \partial _x^{j_1}u\partial _x^{1+j_1}u\right| \psi \; dx + \int \left( \partial _x^{j_1}u\right) ^2|\psi '| \; dx \\\le & {} \int \left( \partial _x^{1+j_1}u\right) ^2\psi \; dx + \int \left( \partial _x^{j_1}u\right) ^2\psi \; dx + \int \left( \partial _x^{j_1}u\right) ^2|\psi '| \; dx. \end{aligned}$$

This completes the proof of Lemma 3. $\square$

We now turn to the proof of Theorem 1. As the argument is translation invariant, we consider only $x_0=0$. Additionally, the estimates are performed for nonlinearity $u\partial _x^3u$; a later remark explains how to control other terms. We invoke constants $c_0,c_1,c_2,\ldots ,$ depending only on the parameters

$$\begin{aligned} c_k=c_k\left( l,T,\varepsilon ,b,\Vert u_0\Vert _{H^s}; \left\| \partial _x^lu_0\right\| _{L^2(x_0,\infty )}; \left\| \partial _x^3u\right\| _{L_T^1L_x^\infty }\right) \end{aligned}$$

(3.6)

whose value may change from line to line. We explicitly record dependence on the parameter $\nu$ using the notation $c(\nu ;d)$, which indicates a constant taking the form of a degree-d polynomial in $\nu$:

$$\begin{aligned} c(\nu ;d) = c_d\nu ^d + \cdots + c_1\nu + c_0. \end{aligned}$$

We first describe the formal calculations and later provide justification using a limiting argument. Let u be a smooth solution of IVP (1.1), differentiate the equation l-times and apply (3.2) with $\phi (x,t)=\chi (x+\nu t;\varepsilon ,b)$. Using properties (2.4) and (2.5) to expand the region of integration in the first term, we arrive at

$$\begin{aligned}&\frac{d}{dt} \int \left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \int \left( \partial _x^{l+2}u\right) ^2 \chi '(x+\nu t) \; dx \nonumber \\&\quad \le \int \left( \partial _x^lu\right) ^2\left\{ \nu \chi '(x+\nu t) + \frac{3}{2}\chi ^{(5)}(x+\nu t) + \frac{25}{16}\frac{\left( \chi '''(x+\nu t)\right) ^2}{\chi '(x+\nu t)}\right\} \; dx \nonumber \\&\qquad +\, 2\int \partial _x^lu\partial _x^l\left( u\partial _x^3u\right) \chi (x+\nu t) \; dx\nonumber \\&\quad \le A+B, \end{aligned}$$

(3.7)

where

$$\begin{aligned} A= & {} \nu \int \left( \partial _x^lu\right) ^2\chi '(x+\nu t) \; dx+c(\varepsilon ;b)\int \left( \partial _x^lu\right) ^2\chi '(x+\nu t;\varepsilon /3,b+\varepsilon ) \; dx,\\ B= & {} 2\int \partial _x^lu\partial _x^l\left( u\partial _x^3u\right) \chi (x+\nu t) \; dx. \end{aligned}$$

We have used the convention that when $\varepsilon$ and b are suppressed, $\chi (x)=\chi (x;\varepsilon ,b)$. The argument proceeds via induction on l where, for fixed l, we integrate (3.7) in time, integrate B by parts and apply a correction to account for the loss of derivatives.

${{Case}\,l=1}$ Integrating in the time interval [0, t] and applying (2.3), we obtain

$$\begin{aligned} \left| \int _0^t A \, d\tau \right| \le c_0(1+\nu ) \int _0^t \int (\partial _xu)^2 \; dxd\tau \le c_0(1+\nu )T\Vert u\Vert _{L_T^\infty H_x^1}^2 \end{aligned}$$

(3.8)

where $0\le t\le T$. After integrating by parts, we find

$$\begin{aligned} B= & {} \int \partial _xu\left( \partial _x^2u\right) ^2 \chi (x+\nu t) \; dx + 3 \int u\left( \partial _x^2u\right) ^2 \chi '(x+\nu t) \; dx\nonumber \\&+ \frac{4}{3} \int (\partial _xu)^3 \chi ''(x+\nu t) \; dx - \int u(\partial _xu)^2\chi '''(x+\nu t) \; dx. \end{aligned}$$

(3.9)

The inequality (2.3) and the Sobolev embedding imply

$$\begin{aligned} \left| \int _0^t B \; d\tau \right|\le & {} c_1(\Vert \partial _xu\Vert _{L_T^\infty L_x^\infty }+\Vert u\Vert _{L_T^\infty L_x^\infty }) \int _0^t\int (\partial _xu)^2 + \left( \partial _x^2u\right) ^2 \;dxd\tau \nonumber \\\le & {} c_1T\Vert u\Vert _{L_T^\infty H_x^2}^3. \end{aligned}$$

(3.10)

Integrating the inequality (3.7) and combining (3.8) and (3.10), we obtain

$$\begin{aligned}&\int (\partial _xu)^2\chi (x+\nu t) \; dx + \int _0^t \int \left( \partial _x^3u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \\&\quad \le \int (\partial _xu_0)^2\chi (x) \; dx + \left| \int _0^t A +B \; d\tau \right| \\&\quad \le c_0\nu + c_1. \end{aligned}$$

As the right-hand side is independent of t, the result follows.

${{Case}\,l=2}$ Similar to the previous case, integrating in the time interval [0, t], we find

$$\begin{aligned} \left| \int _0^t A \; d\tau \right| \le c_0(1+\nu ) \int _0^t \int \left( \partial _x^2u\right) ^2 \; dxd\tau \le c_0(1+\nu )T\Vert u\Vert _{L_T^\infty H_x^2}^2 \end{aligned}$$

(3.11)

where $0\le t\le T$. After integrating by parts, we see

$$\begin{aligned} B= & {} - \int \partial _xu\left( \partial _x^3u\right) ^2 \chi (x+\nu t) \; dx + 3 \int u\left( \partial _x^3u\right) ^2 \chi '(x+\nu t) \; dx\nonumber \\&- \int \partial _xu\left( \partial _x^2u\right) ^2\chi ''(x+\nu t) \; dx - \int u\left( \partial _x^2u\right) ^2\chi '''(x+\nu t) \; dx. \end{aligned}$$

(3.12)

This expression exhibits a loss of derivatives in that the term

$$\begin{aligned} \int \partial _xu \left( \partial _x^3u\right) ^2 \chi (x+\nu t) \; dx \end{aligned}$$

(3.13)

can be controlled neither by the well-posedness theory nor by the $l=1$ case (without the technique introduced in Sect. 7). In [15], Kwon introduced a modified energy to overcome a similar issue. In particular, a smooth solution u to the IVP (1.1) satisfies the following identity:

$$\begin{aligned}&\frac{d}{dt} \int u(\partial _xu)^2\chi \,dx\nonumber \\&\quad = -5\int \partial _xu\left( \partial _x^3u\right) ^2\chi \;dx-5\int u\left( \partial _x^3u\right) ^2\chi '\;dx + \frac{28}{3}\int \left( \partial _x^2u\right) ^3\chi ' \; dx\nonumber \\&\qquad +\, 21\int \partial _xu\left( \partial _x^2u\right) ^2\chi '' \; dx + 5\int u\left( \partial _x^2u\right) ^2\chi ''' \; dx - \frac{10}{3} \int (\partial _xu)^3\chi ^{(4)} \; dx \nonumber \\&\qquad - \int u(\partial _xu)^2\chi ^{(5)} \; dx + 4\int u\partial _xu\left( \partial _x^2u\right) ^2\chi \; dx + 3\int u^2\left( \partial _x^2u\right) ^2\chi ' \; dx \nonumber \\&\qquad - \frac{9}{4}\int (\partial _xu)^4 \chi ' \; dx - \int u\partial _x^2u(\partial _xu)^2\chi ' \; dx - 4\int u(\partial _xu)^3 \chi '' \; dx \nonumber \\&\qquad - \int u^2(\partial _xu)^2 \chi ''' \; dx + \nu \int u(\partial _xu)^2 \chi ' \; dx \end{aligned}$$

(3.14)

where $\chi ^{(j)}$ denotes $\chi ^{(j)}(x+\nu t)$. We use this identity to eliminate (3.13) from (3.12), yielding

$$\begin{aligned} B= & {} \frac{1}{5} \frac{d}{dt}\int u(\partial _xu)^2\chi (x+\nu t) \; dx + 4 \int u\left( \partial _x^3u\right) ^2\chi '(x+\nu t) \; dx\nonumber \\&-\, \frac{4}{5} \int u\partial _xu\left( \partial _x^2u\right) ^2\chi (x+\nu t) \; dx - \frac{\nu }{5} \int u(\partial _xu)^2 \chi '(x+\nu t) \; dx\nonumber \\&+ \sum _{\begin{array}{c} 0 \le j_1,j_2,j_3 \le 2 \\ 1 \le j_4 \le 5 \end{array}}c_{j_1,j_2,j_3,j_{4}} \int \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\left( \partial _x^{j_3}u\right) ^2\chi ^{(j_4)}(x+\nu t) \; dx \end{aligned}$$

(3.15)

where the notation $\widetilde{\partial _x^{j_1}u}$ indicates this factor may be omitted. That is, since $0\le j_1,j_2 \le 2$,

$$\begin{aligned} \left\| \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\right\| _{L_T^\infty L_x^\infty } \le \Vert u\Vert _{L_T^\infty H_x^s} + \Vert u\Vert _{L_T^\infty H_x^s}^2. \end{aligned}$$

Integrating in the time interval [0, t], applying (2.3) and the Sobolev embedding, we obtain

$$\begin{aligned}&\left| \int _0^t\int \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\left( \partial _x^{j_3}u\right) ^2\chi ^{(j_4)}(x+\nu \tau ) \; dxd\tau \right| \nonumber \\&\quad \le c_1\left\| \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\right\| _{L_T^\infty L_x^\infty } \int _0^T\int \left( \partial _x^{j_3}u\right) ^2 \; dxd\tau \nonumber \\&\quad \le c_1T\Vert u\Vert _{L_T^\infty H_x^s}^3(1+\Vert u\Vert _{L_T^\infty H_x^s}) \end{aligned}$$

(3.16)

since $\max \{j_1,j_2,j_3\}\le 2$. The fundamental theorem of calculus and Sobolev embedding yield

$$\begin{aligned} \left| \int _0^t B \; d\tau \right|\le & {} \left| \int u_0(\partial _xu_0)^2 \chi (x)\;dx\right| + \left| \int u(\partial _xu)^2\chi (x+\nu t)\; dx\right| \nonumber \\&+\, 4 \Vert u\Vert _{L_T^\infty H_x^1}\int _0^T\int \left( \partial _x^3u\right) ^2\chi '(x+\nu \tau ) \; dxd\tau \nonumber \\&+\, \frac{4}{5} \Vert u\Vert _{L_T^\infty H_x^2}^2 \int _0^T\int \left( \partial _x^2u\right) ^2\chi (x+\nu \tau ) \; dxd\tau \nonumber \\&+\, \frac{\nu }{5} \Vert u\Vert _{L_T^\infty H_x^1}\int _0^T\int (\partial _xu)^2 \chi '(x+\nu \tau ) \; dxd\tau \nonumber \\&+\, c_1T\Vert u\Vert _{L_T^\infty H_x^s}^3(1+\Vert u\Vert _{L_T^\infty H_x^s}). \end{aligned}$$

(3.17)

The first term on the right-hand side is controlled by the Sobolev embedding, the hypothesis on the initial data and Lemma 2. The second and third term illustrate the iterative nature of the argument, as they can be bounded by the $l=1$ result. The two remaining integrals are finite by property (2.3). Therefore

$$\begin{aligned} \left| \int _0^t B \; d\tau \right| \le c_0\nu +c_1. \end{aligned}$$

(3.18)

Integrating inequality (3.7), using (3.11), (3.18) and the hypothesis on the initial data, we have

$$\begin{aligned}&\int \left( \partial _x^2u\right) ^2\chi (x+\nu t) \; dx + \int _0^t \int \left( \partial _x^4u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \\&\quad \le \int \left( \partial _x^2u_0\right) ^2\chi (x) \; dx + \left| \int _0^t A + B \; d\tau \right| \\&\quad \le c_0\nu +c_1. \end{aligned}$$

As the right-hand side is independent of t, the result follows.

$\underline{\hbox {Case}\,l=3}$ Integrating in the time interval [0, t] and applying the $l=1$ result, we obtain

$$\begin{aligned} \left| \int _0^t A \; d\tau \right|\le & {} \nu \int _0^T \int \left( \partial _x^3u\right) ^2\chi '(x+\nu \tau ) \; dxd\tau \nonumber \\&+\, c_0\int _0^T \int \left( \partial _x^3u\right) ^2\chi '(x+\nu \tau ;\varepsilon /3,b+\varepsilon ) \; dxd\tau \nonumber \\\le & {} c_2\nu ^2 + c_1\nu + c_0 \end{aligned}$$

(3.19)

where $0\le t\le T$. After integrating by parts, we find

$$\begin{aligned} B= & {} -3 \int \partial _xu\left( \partial _x^4u\right) ^2 \chi (x+\nu t) \; dx + 3 \int u\left( \partial _x^4u\right) ^2 \chi '(x+\nu t) \; dx\nonumber \\&+ \int \left( \partial _x^3u\right) ^3\chi (x+\nu t) \; dx - \int u\left( \partial _x^3u\right) ^2\chi '''(x+\nu t) \; dx. \end{aligned}$$

(3.20)

This expression exhibits a loss of derivatives in the term

$$\begin{aligned} \int \partial _xu \left( \partial _x^4u\right) ^2 \chi (x+\nu t) \; dx. \end{aligned}$$

(3.21)

A smooth solution u to the IVP (1.1) satisfies the following identity:

$$\begin{aligned}&\frac{d}{dt} \int u\left( \partial _x^2u\right) ^2\chi \; dx\nonumber \\&\quad = -5\int \partial _xu\left( \partial _x^4u\right) ^2\chi \;dx - 5\int u\left( \partial _x^4u\right) ^2\chi '\;dx \nonumber \\&\quad \qquad +\, 5\int \left( \partial _x^3u\right) ^3\chi \; dx + 25\int \partial _x^2u\left( \partial _x^3u\right) ^2\chi ' \; dx + 15\int \partial _xu\left( \partial _x^3u\right) ^2\chi '' \; dx\nonumber \\&\qquad \quad +\, 5\int u\left( \partial _x^3u\right) ^2\chi ''' \; dx + 2\int u\partial _xu\left( \partial _x^3u\right) ^2\chi \; dx + 3\int u^2\left( \partial _x^3u\right) ^2\chi ' \; dx \nonumber \\&\qquad \quad -\, \frac{25}{3}\int \left( \partial _x^2u\right) ^3\chi ''' \; dx - 5\int \partial _xu\left( \partial _x^2u\right) ^2\chi ^{(4)} \; dx - \int u\left( \partial _x^2u\right) ^2\chi ^{(5)} \; dx \nonumber \\&\qquad \quad -\, \int \partial _xu\left( \partial _x^2u\right) ^3 \chi \; dx - 3\int u\left( \partial _x^2u\right) ^2 \chi ' \; dx - 2\int \left( \partial _xu\right) ^2\left( \partial _x^2u\right) ^2\chi ' \; dx\nonumber \\&\qquad \quad -\, 4\int u\partial _xu\left( \partial _x^2u0^2\chi ''\right. \; dx - \int u^2\left( \partial _x^2u\right) ^2 \chi ''' \; dx + \nu \int u\left( \partial _x^2u\right) ^2\chi ' \; dx\qquad \qquad \end{aligned}$$

(3.22)

where $\chi ^{(j)}$ denotes $\chi ^{(j)}(x+\nu t)$, which we use to eliminate (3.21) from (3.20). Thus, ignoring coefficients, we may write

$$\begin{aligned} B= & {} \frac{d}{dt}\int u\left( \partial _x^2u\right) ^2\chi (x+\nu t) \; dx + \int u\left( \partial _x^4u\right) ^2\chi '(x+\nu t) \; dx \nonumber \\&+ \int \left( 1+u\partial _xu+\partial _x^3u\right) \left( \partial _x^3u\right) ^2\chi (x+\nu t) \; dx + \nu \int u\left( \partial _x^2u\right) ^2\chi ' \; dx \nonumber \\&+ \sum _{\begin{array}{c} 0\le j_1,j_2\le 2 \\ 1 \le j_3 \le 3 \end{array}}c_{j_1,j_2,j_{3}} \int \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\left( \partial _x^3u\right) ^2\chi ^{(j_3)}(x+\nu t)\,dx \nonumber \\&+ \sum _{\begin{array}{c} 0\le j_1,j_2\le 2 \\ 1 \le j_3 \le 5 \end{array}}c_{j_1,j_2,j_{3}} \int \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\left( \partial _x^2u\right) ^2\chi ^{(j_3)}(x+\nu t) \; dx \end{aligned}$$

(3.23)

where the notation $\widetilde{\partial _x^{j_1}u}$ indicates this factor may be omitted. Integrating in the time interval [0, t], applying (2.5), the Sobolev embedding and the $l=1$ result yields

$$\begin{aligned}&\left| \int _0^t\int \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\left( \partial _x^3u\right) ^2\chi ^{(j_3)}(x+\nu \tau ) \; dxd\tau \right| \nonumber \\&\quad \le c_1\left\| \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\right\| _{L_T^\infty L_x^\infty } \int _0^T\int \left( \partial _x^3u\right) ^2\chi '(x+\nu \tau ;\varepsilon /3,b+\varepsilon ) \; dxd\tau \nonumber \\&\quad \le \left( \Vert u\Vert _{L_T^\infty H_x^s}+\Vert u\Vert _{L_T^\infty H_x^s}^2\right) (c_0\nu +c_1). \end{aligned}$$

(3.24)

Similarly, integrating in the time interval [0, t], applying (2.3) and the Sobolev embedding, we find

$$\begin{aligned}&\left| \int _0^t\int \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\left( \partial _x^2u\right) ^2\chi ^{(j_3)}(x+\nu \tau ) \; dxd\tau \right| \nonumber \\&\quad \le c_1\left\| \widetilde{\partial _x^{j_1}u}\partial _x^{j_2}u\right\| _{L_T^\infty L_x^\infty } \int _0^T\int \left( \partial _x^2u\right) ^2 \; dxd\tau \nonumber \\&\quad \le c_1T\Vert u\Vert _{L_T^\infty H_x^s}^3(1+\Vert u\Vert _{L_T^\infty H_x^s}). \end{aligned}$$

(3.25)

Hence the fundamental theorem of calculus and Sobolev embedding yield

$$\begin{aligned} \left| \int _0^t B \; d\tau \right|\le & {} \left| \int u_0\left( \partial _x^2u_0\right) ^2 \chi (x)\;dx\right| + \left| \int u\left( \partial _x^2u\right) ^2\chi (x+\nu t)\; dx\right| \nonumber \\&+\, \Vert u\Vert _{L_T^\infty H_x^1}\int _0^T\int \left( \partial _x^4u\right) ^2\chi '(x+\nu \tau ) \; dxd\tau \nonumber \\&+ \int _0^t \left( 1+\Vert u\Vert _{L_T^\infty H_x^2}^2 + \Vert \partial _x^3u(\tau )\Vert _{L_x^\infty }\right) \int \left( \partial _x^3u\right) ^2\chi (x+\nu \tau ) \; dxd\tau \nonumber \\&+ \left( \Vert u\Vert _{L_T^\infty H_x^s}+\Vert u\Vert _{L_T^\infty H_x^s}^2\right) (c_0\nu +c_1)\nonumber \\&+\, c_1T\Vert u\Vert _{L_T^\infty H_x^s}^3(1+\Vert u\Vert _{L_T^\infty H_x^s}). \end{aligned}$$

(3.26)

Similar to the $l=2$ case, the first term on the right-hand side is controlled by the hypothesis on the initial data. The second and third terms are finite by the $l=2$ case. Therefore

$$\begin{aligned} \left| \int _0^t B \; d\tau \right| \le c(\nu ;1) + \int _0^t \left( c_0+c_1\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) \int \left( \partial _x^3u\right) ^3\chi (x+\nu \tau ) \; dxd\tau .\qquad \quad \end{aligned}$$

(3.27)

Integrating inequality (3.7), using (3.19), (3.27) and the hypothesis on the initial data,

$$\begin{aligned} y(t):= & {} \int \left( \partial _x^3u\right) ^2\chi (x+\nu t) \; dx + \int _0^t \int \left( \partial _x^5u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \\\le & {} \int \left( \partial _x^3u_0\right) ^2\chi (x) \; dx + \left| \int _0^t A + B \; d\tau \right| \\\le & {} c(\nu ;2) + \int _0^t \left( c_0+c_1\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) \int \left( \partial _x^3u\right) ^2\chi (x+\nu \tau ) \; dxd\tau \\\le & {} c(\nu ;2) + \int _0^t \left( c_0+c_1\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) y(\tau ) \; dxd\tau . \end{aligned}$$

Applying Gronwall’s inequality produces

$$\begin{aligned}&\sup _{0\le t \le T} \int \left( \partial _x^4u\right) ^2\chi (x+\nu t) \; dx + \int _0^T\int \left( \partial _x^5u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \\&\quad \le c(\nu ;2)\exp \left( c_0T + c_1\left\| \partial _x^3u\right\| _{L_T^1L_x^\infty }\right) . \end{aligned}$$

This proves the desired result with $l=3$.

${{Cases}\,l=4,5,6}$ Due to the structure of the IVP, the cases $l=4,5,6$ must be handled individually. The analysis is omitted as it is similar to the cases $l=3$ and $l\ge 7$. It can be proved that

$$\begin{aligned} \sup _{0\le t \le T} \int \left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \int _0^T \int \left( \partial _x^{l+2}u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \le c(\nu ;d) \end{aligned}$$

where the values of d are summarized in Table 1.

Table 1 Summary of degree d of $\nu$-dependence of constants for $l=1,2,\ldots ,6$

${{Case}\,l\ge 7}$ In the course of this case, we will prove that for $l\ge 7$, the final constant obtained after integrating both sides of (3.7) takes the form of a polynomial in $\nu$ with degree $8(l-5)$.

Integrating in the time interval [0, t] and applying the $l-2$ result (assuming $l>7$) we have

$$\begin{aligned} \left| \int _0^t A \; d\tau \right|\le & {} \nu \int _0^T \int \left( \partial _x^lu\right) ^2\chi '(x+\nu \tau ) \; dxd\tau \nonumber \\&+\, c_0\int _0^T \int \left( \partial _x^lu\right) ^2\chi '(x+\nu \tau ;\varepsilon /3,b+\varepsilon ) \; dxd\tau \nonumber \\\le & {} c(\nu ;1+8(l-7)) \end{aligned}$$

(3.28)

where $0\le t\le T$. For $l=7$, this expression has degree 5 in $\nu$. We write

$$\begin{aligned} B=B_{1}+B_{2} \end{aligned}$$

(3.29)

where

$$\begin{aligned} B_{1}= & {} 2\int \partial _x^lu \left\{ u\partial _x^{l+3}u +\left( {\begin{array}{c}l\\ 1\end{array}}\right) \partial _xu\partial _x^{l+2}u +\left( {\begin{array}{c}l\\ 2\end{array}}\right) \partial _x^2u\partial _x^{l+1}u\right. \nonumber \\&\left. +\left( 1+\left( {\begin{array}{c}l\\ 3\end{array}}\right) \right) \partial _x^3u\partial _x^lu\right\} \chi (x+\nu t) \; dx\\ B_{2}= & {} \sum _{k=1}^{\lceil {l/2}\rceil -2} c_{l,k}\int \partial _x^{3+k}u\partial _x^{l-k}u\partial _x^lu\chi (x+\nu t) \; dx \end{aligned}$$

and $3+k\le l-k<l$ for $1\le k\le \lceil {l/2}\rceil -2$. Integrating by parts, we have

$$\begin{aligned} B_1=B_{11}+B_{12}, \end{aligned}$$

(3.30)

where

$$\begin{aligned} B_{11}= & {} (3-2l)\int \partial _xu\left( \partial _x^{l+1}u\right) ^2\chi (x+\nu t) \; dx,\\ B_{12}= & {} \int u\left( \partial _x^{l+1}u\right) ^2\chi '(x+\nu t) \; dx +\int \partial _x^3u\left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx\\&+\, \int \partial _x^2u\left( \partial _x^lu\right) ^2\chi '(x+\nu t) \; dx + \int \partial _xu\left( \partial _x^lu\right) ^2\chi ''(x+\nu t) \; dx\\&+\, \int u\left( \partial _x^lu\right) ^2\chi '''(x+\nu t) \; dx \end{aligned}$$

and, in $B_{12}$, we have omitted coefficients depending only on l using the expression (3.30). Then integrating in the time interval [0, t], where $0 \le t \le T$, we obtain

$$\begin{aligned} \left| \int _0^t B_{12} \; d\tau \right|\le & {} \Vert u\Vert _{L_T^\infty H_x^1}\int _0^T\int \left( \partial _x^{l+1}u\right) ^2\chi '(x+\nu \tau ) \; dxd\tau \\&+\, \int _0^t \left\| \partial _x^3u(\tau )\right\| _{L_x^\infty } \int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau ) \; dxd\tau \\&+\, c_0\Vert u\Vert _{L_T^\infty H_x^s}\int _0^t\int \left( \partial _x^lu\right) ^2\chi '(x+\nu \tau ) \; dxd\tau \end{aligned}$$

by the Sobolev embedding and (2.5). Applying the result for cases $l-1$ and $l-2$,

$$\begin{aligned} \left| \int _0^t B_{12} \; d\tau \right| \le c(\nu ;8(l-6)) + \int _0^t \left\| \partial _x^3u(\tau )\right\| _{L_x^\infty } \int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau ) \; dxd\tau .\quad \qquad \end{aligned}$$

(3.31)

Observe that term $B_2$ only occurs when $l\ge 5$. For $l>5$, note that $4+k<l$. The inequality (3.5) produces

$$\begin{aligned} \left| B_2\right|\le & {} \sum _{k=1}^{\lceil {l/2}\rceil -2} c_{l,k} \int \left| \partial _x^{3+k}u\partial _x^{l-k}u\partial _x^lu\right| \chi (x+\nu t)\; dx\nonumber \\\le & {} \int \left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx \nonumber \\&+ \sum _{k=1}^{\lceil {l/2}\rceil -2} \left\{ \int \left( \partial _x^{4+k}u\right) ^2\chi (x+\nu t) \; dx + \int \left( \partial _x^{3+k}u\right) ^2\chi (x+\nu t) \; dx\right. \nonumber \\&\left. + \int \left( \partial _x^{3+k}u\right) ^2\chi '(x+\nu t) \; dx\right\} \int \left( \partial _x^{l-k}u\right) ^2\chi (x+\nu t;\varepsilon /5,4\varepsilon /5) \; dx, \end{aligned}$$

(3.32)

after suppressing constants depending on l. Integrating in the time interval [0, t],

$$\begin{aligned} \left| \int _0^t B_2 \; d\tau \right|\le & {} \int _0^t\int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau ) \; dx \nonumber \\&+\, T\sum _{k=1}^{\lceil {l/2}\rceil -2} \left( \sup _{0 \le t \le T}\int \left( \partial _x^{l-k}u\right) ^2\chi (x+\nu t;\varepsilon /5,4\varepsilon /5) \; dx\right) \\&\times \left( \sup _{0 \le t \le T}\int \left( \partial _x^{4+k}u\right) ^2\chi (x+\nu t) \; dx\right) \\&+\, T\sum _{k=1}^{\lceil {l/2}\rceil -2} \left( \sup _{0 \le t \le T}\int \left( \partial _x^{l-k}u\right) ^2\chi (x+\nu t;\varepsilon /5,4\varepsilon /5) \; dx\right) \\&\times \left( \sup _{0 \le t \le T}\int \left( \partial _x^{3+k}u\right) ^2\chi (x+\nu t) \; dx\right) \\&+\, T\sum _{k=1}^{\lceil {l/2}\rceil -2} \left( \sup _{0 \le t \le T}\int \left( \partial _x^{l-k}u\right) ^2\chi (x+\nu t;\varepsilon /5,4\varepsilon /5) \; dx\right) \\&\times \left( \sup _{0 \le t \le T}\int \left( \partial _x^{3+k}u\right) ^2\chi '(x+\nu t) \; dx\right) . \end{aligned}$$

The strongest $\nu$-dependence for $B_2$ arises from analyzing terms of the form:

$$\begin{aligned} \left( \sup _{0 \le t \le T} \int \left( \partial _x^{l-k}u\right) ^2\chi (x+\nu t;\varepsilon /5,4\varepsilon /5) \; dx \right) \left( \sup _{0 \le t \le T} \int \left( \partial _x^{4+k}u\right) ^2\chi (x+\nu t) \; dx\right) .\nonumber \\ \end{aligned}$$

(3.33)

Each factor in (3.33) is finite by the result for cases $l-k$ and $4+k$. The inductive hypothesis further implies that the $\nu$-dependence has the form of a polynomial in $\nu$ having degree

$$\begin{aligned} \nu ^{8(l-k-5)}\cdot \nu ^{8(4+k-5)} = \nu ^{8(l-6)}. \end{aligned}$$

Hence

$$\begin{aligned} \left| \int _0^t B_2 \; d\tau \right| \le c(\nu ;8(l-6)) + c_0\int _0^t\int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau ) \; dxd\tau . \end{aligned}$$

(3.34)

Integrating the inequality (3.7) in the time interval [0, t], where $0 \le t \le T$, we have

$$\begin{aligned}&\int \left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \int _0^t\int \left( \partial _x^{l+2}u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \nonumber \\&\quad \le \int \left( \partial _x^lu_0\right) ^2\chi (x) \; dx + \left| \int _0^t A + B_{11} + B_{12} + B_2 \; d\tau \right| \nonumber \\&\quad \le c(\nu ;8(l-6))\nonumber \\&\qquad + \left| \int _0^t B_{11} \; d\tau \right| + \int _0^t \left( c_0+c_1\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) \int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau ) \; dxd\tau \qquad \quad \end{aligned}$$

(3.35)

using the hypothesis on the initial data, (3.28), (3.31) and (3.34). Thus it only remains to estimate the integral involving

$$\begin{aligned} B_{11} = (3-2l)\int \partial _xu\left( \partial _x^{l+1}u\right) ^2\chi (x+\nu t) \; dx, \end{aligned}$$

which exhibits a loss of derivatives. Assuming that u satisfies the IVP (1.1), we rewrite this term by considering the correction factor

$$\begin{aligned}&\frac{d}{dt} \int u\left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx \nonumber \\&\quad =\int \partial _x^5u\left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx +\int u\partial _x^3u\left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx \nonumber \\&\qquad +\, 2\int u\partial _x^{l-1}u\partial _x^{l+4}u\chi (x+\nu t) \; dx +2\int u\partial _x^{l-1}u\partial _x^{l-1}\left( u\partial _x^3u\right) \chi (x+\nu t)\, dx\nonumber \\&\qquad +\, \nu \int u\left( \partial _x^{l-1}u\right) ^2\chi '(x+\nu t) \; dx\nonumber \\&\quad =:C_{1}+C_{2}+\widetilde{C_3}+C_{4}+C_{5}. \end{aligned}$$

(3.36)

Observe that integrating $\widetilde{C_3}$ by parts reveals

$$\begin{aligned} \widetilde{C_3} =\left( \frac{5}{2l-3}\right) B_{11}+C_3, \end{aligned}$$

(3.37)

where

$$\begin{aligned} C_3= & {} - 5\int u\left( \partial _x^{l+1}u\right) ^2\chi ' \; dx + 5\int \partial _x^3u\left( \partial _x^lu\right) ^2\chi \; dx\nonumber \\&+\, 9\int \partial _x^2u\left( \partial _x^lu\right) ^2\chi ' \; dx + 15\int \partial _xu\left( \partial _x^lu\right) ^2\chi '' \; dx + \int u\left( \partial _x^lu\right) ^2\chi ''' \; dx\nonumber \\&-\, 5\int \partial _x^5u\left( \partial _x^{l-1}u\right) ^2\chi \; dx - 5\int \partial _x^4u\left( \partial _x^{l-1}u\right) ^2\chi ' \; dx - 9\int \partial _x^3u\left( \partial _x^{l-1}u\right) ^2\chi '' \; dx\nonumber \\&-\, 10\int \partial _x^2u\left( \partial _x^{l-1}u\right) ^2\chi ''' \; dx - 5\int \partial _xu\left( \partial _x^{l-1}u\right) ^2\chi ^{(4)} \; dx - \int u\left( \partial _x^{l-1}u\right) ^2\chi ^{(5)} \; dx.\nonumber \\ \end{aligned}$$

(3.38)

Here $\chi ^{(j)}$ denotes $\chi ^{(j)}(x+\nu t;\varepsilon ,b)$. The fundamental theorem of calculus leads to

$$\begin{aligned} \left( \frac{5}{2l-3}\right) \left| \int _0^t B_{11} \; d\tau \right|\le & {} \left| \int u_0\left( \partial _x^{l-1}u_0\right) ^2\chi (x) \; dx\right| + \left| \int u\left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx\right| \nonumber \\&+ \left| \int _0^t C_1 + C_2 + C_3 + C_4 + C_5 \; d\tau \right| . \end{aligned}$$

(3.39)

We now concern ourselves with estimating the right-hand side of this expression. By the Sobolev embedding, hypothesis on the initial data, Lemma 2 and the result for case $l-1$, we have

$$\begin{aligned}&\left| \int u_0\left( \partial _x^{l-1}u_0\right) ^2\chi (x) \; dx\right| + \left| \int u\left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx\right| \nonumber \\&\quad \le \Vert u_0\Vert _{H^s} \left\| \partial _x^{l-1}u_0\right\| _{L_x^2((0,\infty ))}^2 + \Vert u\Vert _{L_T^\infty H_x^s} \int \left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx, \end{aligned}$$

(3.40)

which is uniformly bounded by the inductive hypothesis. Applying (3.5), we obtain

$$\begin{aligned} |C_1|\le & {} \int \partial _x^5u\left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \;dx\nonumber \\\le & {} \int \left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx \nonumber \\&+\left\{ \int \left( \partial _x^6u\right) ^2\chi (x+\nu t) \; dx {+} \int \left( \partial _x^5u\right) ^2\chi (x+\nu t) dx {+} \int \left( \partial _x^5u\right) ^2\chi '(x+\nu t) \; dx\right\} \nonumber \\&\times \int \left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t;\varepsilon /5,4\varepsilon /5) \; dx. \end{aligned}$$

Integrating in the time interval [0, t] and following the argument applied to term $B_2$, we see that the strongest $\nu$-dependence for $C_1$ arises from analyzing the term

$$\begin{aligned} \left( \sup _{0 \le t \le T} \int \left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t;\varepsilon /5,4\varepsilon /5) \; dx \right) \left( \sup _{0 \le t \le T} \int \left( \partial _x^6u\right) ^2\chi (x+\nu t) \; dx\right) .\nonumber \\ \end{aligned}$$

(3.41)

Each factor in (3.41) is finite by the result for cases 6 and $l-1$. Hence for the base case $l=7$, the right-hand side is bounded by $c(\nu ;16)$. For $l>7$, the inductive hypothesis further yields that the $\nu$-dependence has the form of a polynomial in $\nu$ with degree determined by

$$\begin{aligned} \nu ^{8(l-6)}\cdot \nu ^8 = \nu ^{8(l-5)}. \end{aligned}$$

Thus

$$\begin{aligned} \left| \int _0^t C_1 \; d\tau \right| \le c(\nu ;8(l-5)). \end{aligned}$$

(3.42)

It will be clear from the remainder of the argument that (3.41) produces the overall strongest $\nu$-dependence, hence justifying this inductive calculation.

Integrating in time, using the Sobolev embedding and inductive hypothesis, we find

$$\begin{aligned} \left| \int _0^t C_2 \; d\tau \right|\le & {} \Vert u\Vert _{L_T^\infty H_x^s}\int _0^T\int \left| \partial _x^3u\right| \left( \partial _x^{l-1}u\right) ^2\chi (x+\nu \tau ) \; dxd\tau \nonumber \\\le & {} \Vert u\Vert _{L_T^\infty H_x^s} \int _0^T \left\| \partial _x^3u(\tau )\right\| _{L_x^\infty } \left( \sup _{0 \le t \le T}\int \left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx\right) d\tau \nonumber \\\le & {} c(\nu ;8(l-6))\Vert u\Vert _{L_T^\infty H_x^s} \left\| \partial _x^3u\right\| _{L_T^1L_x^\infty }. \end{aligned}$$

(3.43)

Integrating in time and using (2.5), (3.5), the Sobolev embedding and the inductive hypothesis, we have

$$\begin{aligned} \left| \int _0^t C_3 \; d\tau \right| \le c(\nu ,8(l-6)) + \int _0^t\left( c_0+c_1\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) \int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau ) \; dxd\tau .\nonumber \\ \end{aligned}$$

(3.44)

Expanding but ignoring binomial coeffiecients, we write $C_4=C_{41}+C_{42}$ with

$$\begin{aligned} C_{41}= & {} \int u\partial _xu\left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \int u^2\left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx\nonumber \\&+ \int u\partial _x^3u\left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx + \int \partial _xu\partial _x^2u\left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx\nonumber \\&+ \int u\partial _x^2u\left( \partial _x^{l-1}u\right) ^2\chi '(x+\nu t) \; dx + \int \partial _xu\partial _xu\left( \partial _x^{l-1}u\right) ^2\chi '(x+\nu t) \; dx\nonumber \\&+ \int u\partial _xu\left( \partial _x^{l-1}u\right) ^2\chi ''(x+\nu t) \; dx - \int u^2\left( \partial _x^{l-1}u\right) ^2\chi '''(x+\nu t) \; dx\qquad \end{aligned}$$

(3.45)

and

$$\begin{aligned} C_{42} = \sum _{k=1}^{\lfloor (l-1)/2\rfloor -2} c_{l,k} \int u\partial _x^{(l-1)-k}u\partial _x^{3+k}u\partial _x^{l-1}u\chi (x+\nu t) \; dx. \end{aligned}$$

(3.46)

Similar to $C_2$ and $C_3$,

$$\begin{aligned} \left| \int _0^t C_{41} \; d\tau \right| \le c(\nu ;8(l-6)) + c_0\int _0^t\int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau )\;dxd\tau . \end{aligned}$$

(3.47)

Similar to $B_2$, ignoring constants we have

$$\begin{aligned} \left| \int _0^t C_{42} \; d\tau \right|\le & {} \sum _{k=1}^{\lfloor (l-1)/2\rfloor -2} \int _0^T\int \left| u\partial _x^{(l-1)-k}u\partial _x^{3+k}u\partial _x^{l-1}u\right| \chi \; dxd\tau \nonumber \\\le & {} c(\nu ;8(l-6)) \end{aligned}$$

(3.48)

after applying (3.5). Finally, assuming $l>7$, we obtain

$$\begin{aligned} \left| \int _0^t C_5 \; d\tau \right|\le & {} \nu \Vert u\Vert _{L_T^\infty H_x^{5/2^+}} \int _0^T\int \left( \partial _x^{l-1}u\right) ^2\chi '(x+\nu \tau ) \; dxd\tau \\\le & {} c(\nu ;1+8(l-7)) \end{aligned}$$

(or $c(\nu ;3)$ when $l=7$) using the Sobolev embedding and inductive case $l-3$.

Inserting the above into (3.39) and (3.35), then using nonnegativity of $\chi ,\chi '$, we find

$$\begin{aligned} y(t):= & {} \int \left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \int _0^t\int \left( \partial _x^{l+2}u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \nonumber \\\le & {} c(\nu ;8(l-5)) + \int _0^t\left( c_0+c_1\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) \int \left( \partial _x^lu\right) ^2 \chi (x+\nu \tau ) \; dxd\tau \nonumber \\\le & {} c(\nu ;8(l-5)) + \int _0^t\left( c_0+c_1\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) y(\tau )\;d\tau . \end{aligned}$$

(3.49)

Hence Gronwall’s inequality yields

$$\begin{aligned}&\sup _{0\le t \le T} \int \left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \int _0^T\int \left( \partial _x^{l+2}u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \\&\quad \le c(\nu ;8(l-5))\exp \left( c_0T +c_1\left\| \partial _x^3u\right\| _{L_T^1L_x^\infty }\right) . \end{aligned}$$

This concludes the proof for the case of smooth data.

Now we use a limiting argument to justify the previous computations for arbitrary $u_0 \in H^{s}(\mathbb {R})$ with $s>5/2$. Fix $\rho \in C_0^\infty (\mathbb {R})$ with $\text {supp}\;\rho \subseteq (-1,1)$, $\rho \ge 0$, $\int \rho (x)\;dx=1$ and

$$\begin{aligned} \rho _\mu (x) = \frac{1}{\mu }\rho \left( \frac{x}{\mu }\right) , \quad \mu >0. \end{aligned}$$

The solution $u^\mu$ of IVP (1.1) corresponding to smoothed data $u_0^\mu = \rho _\mu *u_0$, $\mu \ge 0$, satisfies

$$\begin{aligned} u^\mu \in C^{\infty }([0,T] : H^\infty (\mathbb {R})). \end{aligned}$$

Hence we may conclude

$$\begin{aligned} \sup _{0\le t \le T} \int \left( \partial _x^lu^\mu \right) ^2\chi (x+\nu t) \; dx + \int _0^T\int \left( \partial _x^{l+2}u^\mu \right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \le c. \end{aligned}$$

where

$$\begin{aligned} c=c\left( l, \nu , \varepsilon , R, T; \left\| u_0^\mu \right\| _{H^s}; \left\| \partial _x^lu_0^\mu \right\| _{L^2(0,\infty )}; \Vert u^\mu \Vert _{L_T^\infty H_x^s}; \left\| \partial _x^3u^\mu \right\| _{L_T^1L_x^\infty }\right) . \end{aligned}$$

To see that this bound is independent of $\mu >0$, first note

$$\begin{aligned} \left\| u_0^\mu \right\| _{H^s} \le \Vert \widehat{\rho _\mu }\Vert _\infty \Vert u_0\Vert _{H^s} \le \Vert u_0\Vert _{H^s}. \end{aligned}$$

As $\chi \equiv 0$ for $x<\varepsilon$, restricting $0<\mu <\varepsilon$ it follows

$$\begin{aligned} \left( \partial _x^lu_0^\mu \right) ^2\chi (x;\varepsilon ,b) = \left( \rho _\mu *\partial _x^lu_01_{[0,\infty )}\right) ^2\chi (x;\varepsilon ,b). \end{aligned}$$

Thus by Young’s inequality

$$\begin{aligned} \int _\varepsilon ^\infty \left( \partial _x^lu_0^\mu \right) ^2(x) \; dx= & {} \int _\varepsilon ^\infty \left( \rho _\mu *\partial _x^lu_01_{[0,\infty )}\right) ^2(x) \; dx\\\le & {} \Vert \rho _\mu \Vert _1^2 \int _\varepsilon ^\infty \left( \partial _x^lu_0\right) ^2(x)\,dx\\\le & {} \left\| \partial _x^lu_0\right\| _{L^2((0,\infty ))}^2. \end{aligned}$$

From Kwon’s local well-posedness result [15] we have

$$\begin{aligned} \Vert u^\mu \Vert _{L_T^\infty H_x^s} + \left\| \partial _x^{3}u^\mu \right\| _{L_T^1L_x^\infty } \le c\left( \left\| u_0^\mu \right\| _{H^s}\right) \le c(\Vert u_0\Vert _{H^s}) \end{aligned}$$

and so we may replace the bound $c=c(\mu )$ with $\tilde{c}$ as in (1.19).

As the solution depends continuously on the initial data,

$$\begin{aligned} \sup _{0 \le t \le T} \Vert u^\mu (t)-u(t)\Vert _{H^{5/2^+}} \downarrow 0 \quad \text {as}\quad \mu \downarrow 0. \end{aligned}$$

Combining this fact with the $\mu$-uniform bound $\tilde{c}$, weak compactness and Fatou’s lemma, the theorem holds for all $u_0 \in H^{s}(\mathbb {R})$ with $s>5/2$. This completes the proof of Theorem 1 for nonlinearity $u\partial _x^3u$.

Including nonlinearity $\partial _xu\partial _x^2u$, term B in (3.7) will contain a term

$$\begin{aligned} 2\int \partial _x^lu\partial _x^l\left( \partial _xu\partial _x^2u\right) \chi (x+\nu t) \; dx. \end{aligned}$$

As this nonlinearity has a total of three derivatives, integrating by parts produces a form very similar to (3.29). The nonlinearity $u^2\partial _xu$, containing only a single derivative, shows no loss of derivatives (see Sect. 7 for a more thorough treatment). This completes the proof of Theorem 1.

4 Proof of Theorem 2

In this section we prove Theorem 2. Let u be a smooth solution of IVP (1.1), differentiate the equation l-times and apply (3.2) with $\phi (x,t)=\chi _n(x+\nu t;\varepsilon ,b)$ to arrive at

$$\begin{aligned}&\frac{d}{dt} \int \left( \partial _x^lu\right) ^2\chi _n(x+\nu t) \; dx + \int \left( \partial _x^{l+2}u\right) ^2 \chi _n'(x+\nu t) \; dx \nonumber \\&\quad \le A+B, \end{aligned}$$

(4.1)

where

$$\begin{aligned} A= & {} \int \left( \partial _x^lu\right) ^2\left\{ \nu \chi _n'(x+\nu t) + \frac{3}{2}\chi _n^{(5)}(x+\nu t) + \frac{25}{16}\frac{\left( \chi _n'''(x+\nu t)\right) ^2}{\chi _n'(x+\nu t)}\right\} \; dx,\nonumber \\ B= & {} 2\int \partial _x^lu\partial _x^l\left( u\partial _x^3u\right) \chi _n(x+\nu t) \; dx. \end{aligned}$$

The proof proceeds by induction on l, however, for fixed l we induct on n. The base case $n=0$ coincides with the propagation of regularity result. We invoke constants $c_0,c_1,c_2,\ldots ,$ depending only on the parameters

$$\begin{aligned} c_k = c_k\left( n,l; \Vert u_0\Vert _{H^s}; \left\| \partial _x^3u\right\| _{L_T^1L_x^\infty }; \nu ; \varepsilon ; b; T\right) \end{aligned}$$

(4.2)

as well as the decay assumptions on the initial data (1.22).

${{Case}\,l=0}$ Using properties (2.8) and (2.9), we see

$$\begin{aligned} |A| \le c_0\int u^2(1+\chi _n(x+\nu t)) \; dx. \end{aligned}$$

and so integrating in the time interval [0, t], we have

$$\begin{aligned} \left| \int _0^t A \; d\tau \right| \le c_0\left\{ T\Vert u\Vert _{L_T^\infty L_x^2}^2 + \int _0^t \int u^2\chi _n(x+\nu \tau ) \; dxd\tau \right\} \end{aligned}$$

(4.3)

where $0\le t\le T$. Additionally,

$$\begin{aligned} \left| \int _0^t B \; d\tau \right| \le 2\int _0^t\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty } \int u^2\chi _n(x+\nu \tau ) \; dxd\tau . \end{aligned}$$

(4.4)

Integrating (4.1) in the time interval [0, t], combining (4.3) and (4.4), we have

$$\begin{aligned} y(t):= & {} \int u^2\chi _n(x+\nu t) \; dx + \int _0^t\int \left( \partial _x^2u\right) ^2\chi _n(x+\nu \tau )\;dxd\tau \\\le & {} \int u_0^2(x)\chi _n(x) \; dx + \left| \int _0^t A+B \; d\tau \right| \\\le & {} c_0 + \int _0^t \left( c_1 + c_2\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) \int u^2\chi _n(x+\nu \tau ) \; dxd\tau \\\le & {} c_0 + \int _0^t \left( c_1 + c_2\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) y(\tau ) \; dxd\tau . \end{aligned}$$

using the hypothesis on the initial data. Gronwall’s inequality yields

$$\begin{aligned}&\sup _{0 \le t \le T} \int u^2\chi _n(x+\nu t) \; dx + \int _0^T\int \left( \partial _x^2u\right) ^2\chi _n(x+\nu \tau )\;dxd\tau \\&\quad \le c_0\exp \left( c_1T+c_2\left\| \partial _x^3u\right\| _{L_T^1L_x^\infty }\right) . \end{aligned}$$

Note that induction in n was not required in this case.

${{Case}\,l=1}$ Using properties (2.8) and (2.9), we have

$$\begin{aligned} |A| \le c_0\int \left( \partial _xu\right) ^2(1+\chi _n(x+\nu t)) \; dx. \end{aligned}$$

and so integrating in the time interval [0, t], we find

$$\begin{aligned} \left| \int _0^t A \; d\tau \right| \le c_0\left\{ T\Vert u\Vert _{L_T^\infty H_x^1}^2 + \int _0^t \int (\partial _xu)^2\chi _n(x+\nu \tau ) \; dxd\tau \right\} \end{aligned}$$

(4.5)

where $0\le t\le T$. After integrating by parts, we find

$$\begin{aligned} B= & {} \int \partial _xu\left( \partial _x^2u\right) ^2 \chi _n(x+\nu t) \; dx + 3 \int u\left( \partial _x^2u\right) ^2 \chi _n'(x+\nu t) \; dx\nonumber \\&+\, \frac{4}{3} \int (\partial _xu)^3 \chi _n''(x+\nu t) \; dx - \int u(\partial _xu)^2\chi _n'''(x+\nu t) \; dx. \end{aligned}$$

(4.6)

This expression exhibits a loss of derivatives requiring a correction. A smooth solution u to the IVP (1.1) satisfies the following identity

$$\begin{aligned} \frac{d}{dt} \int u^3\chi _n \; dx= & {} -15\int \partial _xu\left( \partial _x^2u\right) ^2\chi _n\;dx -9\int u\left( \partial _x^2u\right) ^2\chi _n'\;dx \nonumber \\&+\, 10\int (\partial _xu)^3\chi _n'' \; dx + 12\int u(\partial _xu)^2\chi _n''' \; dx - \int u^3\chi _n^{(5)} \; dx \nonumber \\&+\, 9\int u(\partial _xu)^3\chi _n \; dx + \frac{27}{2} \int u^2(\partial _xu)^2 \chi _n' \; dx - \frac{3}{4} \int u^4\chi _n''' \; dx \nonumber \\&+\,\nu \int u^3 \chi _n' \; dx \end{aligned}$$

(4.7)

after integrating by parts, where $\chi _n^{(j)}$ denotes $\chi _n^{(j)}(x+\nu t)$. Substituting (4.7), we can write (4.6) as a linear combination of the following terms

$$\begin{aligned} B= & {} \frac{d}{dt} \int u^3 \chi _n \; dx + \int u\left( \partial _x^2u\right) ^2 \chi _n' \; dx \nonumber \\&+ \int (\partial _xu)^3 \chi _n'' \; dx + \int u(\partial _xu)^2 \chi _n''' \; dx + \int u^3 \chi _n^{(5)} \; dx \nonumber \\&+ \int u(\partial _xu)^3 \chi _n \; dx + \int u^2(\partial _xu)^2 \chi _n' \; dx + \int u^4 \chi _n''' \; dx \nonumber \\&+ \,\nu \int u^3 \chi _n' \; dx \nonumber \\=: & {} B_1 + \cdots + B_9. \end{aligned}$$

(4.8)

The fundamental theorem of calculus and the Sobolev embedding yield

$$\begin{aligned} \left| \int _0^t B_1 \; d\tau \right| \le \Vert u_0\Vert _{H^1} \int u_0^2(x) \chi _n(x) \; dx + \Vert u\Vert _{L_T^\infty H_x^1} \int u^2 \chi _n(x+\nu t) \; dx \end{aligned}$$

(4.9)

where $0 \le t \le T$. This term is finite by hypothesis (1.22) and the case $l=0$. Next,

$$\begin{aligned} \left| \int _0^t B_2 \; d\tau \right| \le \Vert u\Vert _{L_T^\infty H_x^1} \int _0^T\int \left( \partial _x^2u\right) ^2 \chi _n'(x+\nu \tau ) \; dxd\tau , \end{aligned}$$

(4.10)

which is finite by case $l=0$. Using (2.11) and the Sobolev embedding, we obtain

$$\begin{aligned}&\left| \int _0^t B_3 + B_4 + B_5 \; d\tau \right| \nonumber \\&\quad \le \Vert u\Vert _{L_T^\infty H_x^2} \int _0^T\int (\partial _xu)^2\left| \chi _n''(x+\nu \tau )\right| + (\partial _xu)^2\left| \chi _n'''(x+\nu \tau )\right| \; dxd\tau \nonumber \\&\qquad + \Vert u\Vert _{L_T^\infty H_x^1} \int _0^T\int u^2\left| \chi _n^{(5)}(x+\nu \tau )\right| \; dxd\tau \nonumber \\&\quad \le c_0\Vert u\Vert _{L_T^\infty H_x^2} \int _0^T\int (\partial _xu)^2\chi _{n-1}(x+\nu \tau ;\varepsilon /3,b+\varepsilon ) \; dxd\tau \nonumber \\&\qquad + c_1\Vert u\Vert _{L_T^\infty H_x^1} \int _0^T\int u^2\chi _{n-1}(x+\nu \tau ;\varepsilon /3,b+\varepsilon ) \; dxd\tau . \end{aligned}$$

(4.11)

The first term is finite by induction on n in the current case $l=1$, whereas the second term is finite by the case $l=0$. The Sobolev embedding implies

$$\begin{aligned} \left| \int _0^t B_6 \; d\tau \right| \le \Vert u\Vert _{L_t^\infty H_x^2}^2 \int _0^t\int (\partial _xu)^2 \chi _n(x+\nu \tau ) \; dxd\tau . \end{aligned}$$

(4.12)

Finally the inequality (2.11) and the Sobolev embedding yield

$$\begin{aligned} \left| \int _0^t B_7 + B_8 + B_9 \; d\tau \right| \le c_2\Vert u\Vert _{L_T^\infty H_x^2}^2 \int _0^T\int u^2\chi _{n-1}(x+\nu \tau ;\varepsilon /3,b+\varepsilon ) \; dxd\tau ,\nonumber \\ \end{aligned}$$

(4.13)

which is finite by case $l=0$. Integrating (4.1) in the time interval [0, t] and combining the above, we have

$$\begin{aligned} y(t):= & {} \int (\partial _xu)^2 \chi _n(x+\nu t) \; dx + \int _0^t\int \left( \partial _x^3u\right) ^2 \chi _n'(x+\nu \tau ) \; dxd\tau \\\le & {} \int (\partial _xu_0)^2(x)\chi _n(x) \; dx + \left| \int _0^t A+B \; d\tau \right| \\\le & {} c_0 + c_1 \int _0^t \int (\partial _xu)^2 \chi _n(x+\nu \tau ) \; dxd\tau \\\le & {} c_0 + c_1 \int _0^t y(\tau ) \; d\tau . \end{aligned}$$

The result follows by Gronwall’s inequality.

${{Cases}\,l=2,3,4,5}$ Due to the structure of the IVP, the cases $l=2,3,4,5$ must be handled individually. The analysis is omitted, however, as it is similar to the cases presented.

${{Case}\,l\ge 6}$ Integrating in the time interval [0, t] and using properties (2.10) and (2.11), we have

$$\begin{aligned} \left| \int _0^t A \; d\tau \right| \le c_0\int _0^t \int \left( \partial _x^lu\right) ^2 \chi _{n-1}(x+\nu \tau ;\varepsilon /3,b+\varepsilon ) \; dxd\tau , \end{aligned}$$

(4.14)

which is finite by induction on n. Recall (3.29) and (3.30), wherein we wrote

$$\begin{aligned} B = B_{11} + B_{12} + B_2, \end{aligned}$$

with the term $B_{11}$ exhibiting a loss of derivatives. Integrating in the time interval [0, t], we see

$$\begin{aligned} \left| \int _0^t B_{12} \; d\tau \right|\le & {} \Vert u\Vert _{L_T^\infty H_x^1} \int _0^T\int \left( \partial _x^{l+1}u\right) ^2 \chi _n'(x+\nu \tau ) \; dxd\tau \nonumber \\&+ \int _0^t \left\| \partial _x^3u(\tau )\right\| _{L_x^\infty } \int \left( \partial _x^lu\right) ^2\chi _n(x+\nu \tau ) \; dxd\tau \nonumber \\&+ \,c_0\Vert u\Vert _{L_T^\infty H_x^s} \int _0^T\int \left( \partial _x^lu\right) ^2 \chi _{n-1}(x+\nu \tau ) \; dxd\tau \end{aligned}$$

(4.15)

where we have used (2.11). The first term is finite by the case $l-1$ and the third is finite by induction on n, hence

$$\begin{aligned} \left| \int _0^t B_{12} \; d\tau \right| \le c_0 + c_1 \int _0^t \left\| \partial _x^3u(\tau )\right\| _{L_x^\infty } \int \left( \partial _x^lu\right) ^2\chi _n(x+\nu \tau ) \; dxd\tau \end{aligned}$$

Observe that term $B_2$ only occurs when $l\ge 5$. For $l>5$, note that $4+k<l$. The inequality (3.5) yields

$$\begin{aligned} |B_2|\le & {} \sum _{k=1}^{\lceil {l/2}\rceil -2} c_{l,k} \int \left| \partial _x^{3+k}u\partial _x^{l-k}u\partial _x^lu\right| \chi _n(x+\nu t)\; dx\nonumber \\\le & {} \int \left( \partial _x^lu\right) ^2\chi _n(x+\nu t) \; dx \nonumber \\&+ \sum _{k=1}^{\lceil {l/2}\rceil -2} \left\{ \int \left( \partial _x^{4+k}u\right) ^2\chi _n(x+\nu t) \; dx + \int \left( \partial _x^{3+k}u\right) ^2\chi _n(x+\nu t) \; dx\right. \nonumber \\&+\left. \int \left( \partial _x^{3+k}u\right) ^2\chi _n'(x+\nu t) \; dx\right\} \int \left( \partial _x^{l-k}u\right) ^2\chi _n(x+\nu t;\varepsilon /5,4\varepsilon /5) \; dx,\nonumber \\ \end{aligned}$$

(4.16)

where we have suppressed constants depending on l. Integrating in the time interval [0, t], we see

$$\begin{aligned} \left| \int _0^t B_2 \; d\tau \right| \le c_0 + c_1\int _0^t\int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau ) \; dxd\tau , \end{aligned}$$

(4.17)

as factors in the summation are estimated via (2.11) and the inductive hypothesis.

Assuming that u satisfies the IVP (1.1), we rewrite this term by considering the correction factor

$$\begin{aligned} \frac{d}{dt} \int u\left( \partial _x^{l-1}u\right) ^2\chi _n(x+\nu t) \; dx =\widetilde{C_1}+C_{2}+C_{3}+C_{4}, \end{aligned}$$

where

$$\begin{aligned} \widetilde{C_1}= & {} \int \partial _x^5u\left( \partial _x^{l-1}u\right) ^2\chi _n(x+\nu t) \; dx + 2\int u\partial _x^{l-1}u\partial _x^{l+4}u\chi _n(x+\nu t) \; dx,\\ C_{2}= & {} \int u\partial _x^3u\left( \partial _x^{l-1}u\right) ^2\chi _n(x+\nu t) \; dx,\\ C_{3}= & {} 2\int u\partial _x^{l-1}u\partial _x^{l-1}\left( u\partial _x^3u\right) \chi _n(x+\nu t) \; dx,\\ C_{4}= & {} \nu \int u\left( \partial _x^{l-1}u\right) ^2\chi _n'(x+\nu t) \; dx. \end{aligned}$$

Integrating $\widetilde{C_1}$ by parts, we have

$$\begin{aligned} \widetilde{C_1} = \left( \frac{5}{2l-3}\right) B_{11}+C_1, \end{aligned}$$

(4.18)

where

$$\begin{aligned} C_1= & {} - 5\int u\left( \partial _x^{l+1}u\right) ^2\chi _n' \; dx + 5\int \partial _x^3u\left( \partial _x^lu\right) ^2\chi _n \; dx \nonumber \\&+\, 15\int \partial _x^2u\left( \partial _x^lu\right) ^2\chi _n' \; dx + 15\int \partial _xu\left( \partial _x^lu\right) ^2\chi _n'' \; dx\nonumber \\&+\, 5\int u\left( \partial _x^lu\right) ^2\chi _n''' \; dx - 5\int \partial _x^4u\left( \partial _x^{l-1}u\right) ^2\chi _n' \; dx\nonumber \\&-\, 10\int \partial _x^3u\left( \partial _x^{l-1}u\right) ^2\chi _n'' \; dx - 10\int \partial _x^2u\left( \partial _x^{l-1}u\right) ^2\chi _n''' \; dx \nonumber \\&-\, 5\int \partial _xu\left( \partial _x^{l-1}u\right) ^2\chi _n^{(4)} \; dx - \int u\left( \partial _x^{l-1}u\right) ^2\chi _n^{(5)} \; dx. \end{aligned}$$

(4.19)

Here $\chi _n^{(j)}$ denotes $\chi _n^{(j)}(x+\nu t;\varepsilon ,b)$. The fundamental theorem of calculus yields

$$\begin{aligned}&\left( \frac{5}{2l-3}\right) \left| \int _0^t B_{11} \; d\tau \right| \\&\quad \le \left| \int u_0\left( \partial _x^{l-1}u_0\right) ^2\chi _n(x) \; dx\right| + \left| \int u\left( \partial _x^{l-1}u\right) ^2\chi _n(x+\nu t) \; dx\right| \\&\qquad + \left| \int _0^t C_1 + C_2 + C_3 + C_4 \; d\tau \right| . \end{aligned}$$

We now concern ourselves with estimating the right-hand side of this expression. First note

$$\begin{aligned}&\left| \int u_0\left( \partial _x^{l-1}u_0\right) ^2\chi _n(x) \; dx\right| + \left| \int u\left( \partial _x^{l-1}u\right) ^2\chi _n(x+\nu t) \; dx\right| \nonumber \\&\quad \le \Vert u_0\Vert _{H^1} \left\| x^{n/2}\partial _x^{l-1}u_0\right\| _{L_x^2(\varepsilon ,\infty )}^2 + \Vert u\Vert _{L_T^\infty H_x^1} \int \left( \partial _x^{l-1}u\right) ^2\chi _n(x+\nu t) \; dx,\qquad \quad \end{aligned}$$

(4.20)

is bounded by the hypothesis (1.22) and the case $l-1$. Similarly to $B_2$ and $B_{12}$, integrating in the time interval [0, t], using (3.5) and property (2.11), we obtain

$$\begin{aligned} \left| \int _0^t C_1 \; d\tau \right| \le c_0 + \int _0^t \left( c_1+c_2\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) \int \left( \partial _x^lu\right) ^2 \chi _n(x+\nu \tau ) \; dxd\tau \qquad \quad \end{aligned}$$

(4.21)

where the term containing $(\partial _x^{l+1}u)^2\chi _n'$ is controlled using the induction case $l-1$, as in (4.15).

Using (3.5) and the inductive hypothesis, we see

$$\begin{aligned} \left| \int _0^t C_2 \; d\tau \right| \le c_0, \end{aligned}$$

(4.22)

similar to $B_2$. The same technique applies to $C_3$ and $C_4$.

Integrating (4.1) in the time interval [0, t] and combining the above, we find that there exists constants as in (4.2) such that

$$\begin{aligned} y(t):= & {} \int \left( \partial _x^lu\right) ^2 \chi _n(x+\nu t) \; dx + \int _0^t\int \left( \partial _x^{l+2}u\right) ^2 \chi _n'(x+\nu \tau ) \; dxd\tau \\\le & {} \int \left( \partial _x^lu_0\right) ^2(x)\chi _n(x) \; dx + \left| \int _0^t A+B \; d\tau \right| \\\le & {} c_0 + \int _0^t \left( c_1+c_2\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) \int \left( \partial _x^lu\right) ^2 \chi _n(x+\nu \tau ) \; dxd\tau \\\le & {} c_0 + \int _0^t \left( c_1+c_2\left\| \partial _x^3u(\tau )\right\| _{L_x^\infty }\right) y(\tau ) \; d\tau . \end{aligned}$$

The result follows by Gronwall’s inequality. To handle the case of arbitrary data $u_0 \in H^{s}(\mathbb {R})$ with $s>5/2$, a limiting argument similar to the proof of Theorem 1 is used. This completes the proof of Theorem 2.

5 Proof of Theorem 3

In this section we prove Theorem 3. Integration by parts yields the next lemma.

Lemma 4

Suppose for some $l\in \mathbb {Z}^+$

$$\begin{aligned} \sup _{0 \le t \le T} \int \left( \partial _x^lu\right) ^2 \chi _n(x+\nu t) \; dx + \int _0^T\int \left( \partial _x^{l+2}u\right) ^2 \chi _n'(x+\nu \tau ) \; dxd\tau < \infty . \end{aligned}$$

(5.1)

Then for every $0<\delta <T$, there exists $\hat{t}\in (0,\delta )$ such that

$$\begin{aligned} \int \left( \partial _x^{l+j}u\right) ^2 \chi _{n-1}\left( x+\nu \hat{t};\varepsilon ^+,b\right) \; dx < \infty \qquad (j=0,1,2). \end{aligned}$$

(5.2)

To prove Theorem 3, it suffices to consider an example; fix $n=9$ in the hypothesis of the theorem. Then we may apply Theorem 2 with $(l,n)=(0,9)$. Thus, after applying Lemma 4, there exists $t_0\in (0,\delta /2)$ such that

$$\begin{aligned} \int \left( u^2 + (\partial _xu)^2 + \left( \partial _x^2u\right) ^2\right) \chi _8(x+\nu t_0;\varepsilon ^+,b) \; dx < \infty . \end{aligned}$$

Hence we may apply Theorem 2 with $(l,n)=(2,8)$ and find $t_1 \in (t_0,\delta /2)$ such that

$$\begin{aligned} \int \left( u^2 + \cdots + \left( \partial _x^4u\right) ^2\right) \chi _7(x+\nu t_1;\varepsilon ^+,b) \; dx < \infty . \end{aligned}$$

Continuing in this manner, applying Theorem 2 with $(l,n)=(4,7),(6,6),\ldots ,(18,0)$ provides the existince of $\hat{t}\in (\delta /2,\delta )$ such that

$$\begin{aligned} \int \left( u^2 + \cdots + \left( \partial _x^{19}u\right) ^2\right) \chi (x+\nu \hat{t};\varepsilon ^+,b) \; dx < \infty . \end{aligned}$$

Finally, we can apply Theorem 1 with $l=19$, completing the proof.

6 Proof of Corollary 2

The proof of Corollary 2 relies on the following lemma, which follows by considering a dyadic decomposition of the interval $[0,\infty )$. Observe that the lemma also applies when integrating a nonnegative function on the interval $[-(a+\varepsilon ),-\varepsilon ]$, implying decay on the left half-line.

Lemma 5

Let $f:[0,\infty ) \rightarrow [0,\infty )$ be continuous. If for $a>0$

$$\begin{aligned} \int _0^a f(x) \; dx \le ca^\alpha \end{aligned}$$

then for every $\varepsilon >0$

$$\begin{aligned} \int _0^\infty \frac{1}{\langle x \rangle ^{\alpha +\varepsilon }} f(x) \; dx \le c(\alpha ,\varepsilon ). \end{aligned}$$

Now we prove Corollary 2.

Proof

Recall that for $l\ge 6$, Theorem 1 with $x_0=0$ states

$$\begin{aligned} \sup _{0 \le t \le T} \int _{\varepsilon -\nu t}^\infty \left( \partial _x^lu\right) ^2(x,t) \; dx \le c(\nu ;8(l-5)). \end{aligned}$$

For fixed $t\in (0,T)$

$$\begin{aligned} \int _{\varepsilon -\nu t}^\infty \left( \partial _x^lu\right) ^2(x,t) \; dx = \left( \int _{\varepsilon -\nu t}^\varepsilon + \int _\varepsilon ^\infty \right) \left( \partial _x^lu\right) ^2(x,t) \; dx := I + II. \end{aligned}$$

Theorem 1 with $\nu =0$ yields control of II, so we focus on I. For $\nu ^*$ large enough, $\nu >\nu ^*$ implies

$$\begin{aligned} I = \int _{\varepsilon -\nu t}^\varepsilon \left( \partial _x^lu\right) ^2(x,t) \; dx \le ct^{-8(l-5)}(\nu t)^{8(l-5)}. \end{aligned}$$

Applying Lemma 5 with $a=\nu t$ and $\alpha =8(l-5)$, we find

$$\begin{aligned} \int _{-\infty }^\varepsilon \frac{1}{\langle x \rangle ^{8(l-5)+\varepsilon }} \left( \partial _x^lu\right) ^2(x,t) \; dx < \infty \end{aligned}$$

for $\varepsilon >0$. This completes the proof of Corollary 2. $\square$

7 Extensions to Other Models

In this section we prove the following extension of Theorem 1, which applies to those equations described by Theorem A.

Theorem 4

Consider the class of initial value problems

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _tu - \partial _x^5u + Q\left( u,\partial _xu,\partial _x^2u,\partial _x^3u\right) = 0, \qquad x,t\in \mathbb {R}, \\ u(x,0) = u_0(x), \end{array}\right. } \end{aligned}$$

(7.1)

where $Q:\mathbb {R}^4\rightarrow \mathbb {R}$ is a polynomial having no constant or linear terms. Let u be a solution to IVP (7.1) satisfying

$$\begin{aligned} u \in C([-T,T] ; X_{s,m}), \quad m\in \mathbb {Z}, s\in \mathbb {R}, \end{aligned}$$

such that $m \ge m_0$ and $s \ge \max \{s_0,2m\}$ for a nonnegative integer $m_0$ and positive real number $s_0$ determined by the form of the nonlinearity Q. If $u_0 \in X_{s,m}$ additionally satisfies

$$\begin{aligned} \left\| \partial _x^lu_0\right\| _{L^2(x_0,\infty )}^2 = \int _{x_0}^\infty \left( \partial _x^lu_0\right) ^2(x) \; dx < \infty , \end{aligned}$$

(7.2)

for some $l \in \mathbb {Z}^+, x_0 \in \mathbb {R}$, then u satisfies

$$\begin{aligned} \sup _{0 \le t \le T} \int _{x_0 + \varepsilon - \nu t}^\infty \left( \partial _x^k u\right) ^2(x,t) \; dx \le c \end{aligned}$$

(7.3)

for any $\nu \ge 0, \varepsilon >0$ and each $k=0,1,\ldots ,l$ with

$$\begin{aligned} c = c\left( l; \nu ; \varepsilon ; T; \Vert u_0\Vert _{X_{s,m}}; \left\| \partial _x^lu_0\right\| _{L^2(x_0,\infty )}\right) . \end{aligned}$$

(7.4)

Moreover, for any $\nu \ge 0, \varepsilon >0$ and $R>\varepsilon$

$$\begin{aligned} \int _0^T \int _{x_0 + \varepsilon - \nu t}^{x_0 + R - \nu t} \left( \partial _x^{l+2}u\right) ^2(x,t) \; dxdt \le \tilde{c} \end{aligned}$$

(7.5)

with

$$\begin{aligned} \tilde{c} = \tilde{c}\left( l; \nu ; \varepsilon ; R; T; \Vert u_0\Vert _{X_{s,m}}; \left\| \partial _x^lu_0\right\| _{L^2(x_0,\infty )}\right) . \end{aligned}$$

(7.6)

Remark 5

Due to the similarities in the proof technique, the comments in this section can be modified to prove extensions of Theorems 2 and 3 to the class (7.1).

Remark 6

Establishing local well-posedness of the IVP (7.1) in the weighted Sobolev spaces $X_{s,m}$ imposes minimum values on m and s, see for instance the contraction principle technique used by Kenig et al. in [12] and [11]. Thus the values of $m_0$ and $s_0$ are determined by considering both the local well-posedness as well as our proof of the propagation of regularity. As we see below, these considerations may differ.

Remark 7

A slight modification to the energy inequality (3.2) allows one to loosen the restriction that Q not contain any linear terms. In particular, the theorem applies to the model (1.6) when coupled with an appropriate local well-posedness theorem. Provided suitable cutoff functions exist, modifications to (3.2) also extend the technique to a class of higher order equations containing the KdV heirarchy.

Proof

Though not strictly necessary, we break the proof into cases based on the form of the nonlinearity Q(u). We treat the case $x_0=0$ as the argument is translation invariant. Following the proof of Theorem 1, let u be a smooth solution of the IVP (7.1). Differentiating the equation l-times, applying (3.2) and using properties of $\chi$, we arrive at

$$\begin{aligned}&\frac{d}{dt} \int \left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \int \left( \partial _x^{l+2}u\right) ^2 \chi '(x+\nu t) \; dx \nonumber \\&\quad \lesssim \int \left( \partial _x^lu\right) ^2\chi '(x+\nu t;\varepsilon /3,b+\varepsilon ) \; dx + \int \partial _x^lu\partial _x^lQ(u)\chi (x+\nu t) \; dx \nonumber \\&\quad =: A+B \end{aligned}$$

(7.7)

The proof proceeds by induction on $l\in \mathbb {Z}^+$. For a given nonlinearity Q(u), there exists $l_0\in \mathbb {Z}^+$ such that the cases $l=0,1,\ldots ,l_0$ can be proved by choosing $s_0$ large enough. Thus it suffices to prove only the inductive step. We describe the formal calculations, omitting the limiting argument.

Integrating in the time interval [0, t] and applying the $l-2$ result we have

$$\begin{aligned} \left| \int _0^t A \; d\tau \right| \le c(\nu ;\varepsilon ;b)\int _0^T \int \left( \partial _x^lu\right) ^2\chi '(x+\nu \tau ) \; dxd\tau \le c_0 \end{aligned}$$

(7.8)

where $0\le t\le T$ and $c_0$ as in (7.4). We now turn to term B.

Case 1 Suppose Q is independent of both $\partial _x^2u$ and $\partial _x^3u$. Then there exists $N\in \mathbb {Z}^+$ such that, after integrating by parts, B is a linear combination of terms of the form

$$\begin{aligned} \int u^{j_0}(\partial _xu)^{j_1}\left( \partial _x^2u\right) ^{j_2}\left( \partial _x^lu\right) ^2 \chi (x+\nu t) \; dx, \qquad j_0,j_1,j_2 \le N, \end{aligned}$$

and

$$\begin{aligned} \int u^{j_0}(\partial _xu)^{j_1}\left( \partial _x^2u\right) ^{j_2}\left( \partial _x^ku\right) ^2 \chi ^{(j_3)}(x+\nu t) \; dx, \qquad j_0,j_1,j_2 \le N \end{aligned}$$

where $1 \le j_3 \le 5$ and $3 \le k \le l+1$. Hence no loss of derivatives occurs. Integrating in the time interval [0, t], applying the induction hypothesis and the Sobolev embedding

$$\begin{aligned} \left| \int _0^t B \; d\tau \right| \le c_0 + c_1 \int _0^t\int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau ) \; dxd\tau \end{aligned}$$

provided $s_0>7/2$, with $c_0$ and $c_1$ as in (7.4). Combining with (7.8), after integrating (7.7) in time and using the hypothesis on the initial data we have

$$\begin{aligned} y(t):= & {} \int \left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \int _0^t\int \left( \partial _x^{l+2}u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \nonumber \\\le & {} c_0 + c_1\int _0^t\int \left( \partial _x^lu\right) ^2\chi (x+\nu \tau ) \; dxd\tau \nonumber \\\le & {} c_0 + c_1\int _0^ty(\tau )\;d\tau . \end{aligned}$$

(7.9)

The result follows by an application of Gronwall’s inequality. The value of $m_0$ is determined by the LWP theory.

Case 2 Suppose Q is a linear combination of quadratic terms (with the exception of $u\partial _x^2u$). After integrating by parts B is a linear combination of terms of the form

$$\begin{aligned} \int \partial _x^ju\left( \partial _x^{l+1}u\right) ^2 \chi (x+\nu t) \; dx, \qquad 1 \le j \le 4 \end{aligned}$$

as well as lower order terms. The correction technique of Theorem 1 can be modified to account for this loss of derivatives. For example, if $Q(u)=\partial _x^2u\partial _x^3u$, then integrating by parts and supressing coefficients

$$\begin{aligned} B = \int \partial _x^2u\left( \partial _x^{l+1}u\right) ^2 \chi (x+\nu t) \; dx + \int \partial _x^4u\left( \partial _x^lu\right) ^2 \chi (x+\nu t) \; dx + \tilde{B} \end{aligned}$$

where $\tilde{B}$ is controlled by induction. For the second term, we impose $s_0>9/2$ to control $\Vert \partial _x^4u\Vert _{L_x^\infty }$. For the first term, consider the correction

$$\begin{aligned} \frac{d}{dt} \int \partial _xu\left( \partial _x^{l-1}u\right) ^2\chi (x+\nu t) \; dx. \end{aligned}$$

In general, more than one correction may be necessary. The remainder of the proof is similar to Theorem 1, thus the value of $m_0$ is determined by the LWP theory. Note that if Q additionally contained higher degree terms independent of $\partial _x^2u$ and $\partial _x^3u$, the above argument applies. Equations in the class (1.1) are of this form.

Case 3 The remaining nonlinearities in the class (7.1) exhibit a loss of derivatives which, in general, cannot be controlled by the correction technique. We illustrate the argument in this case by focusing on the example equation

$$\begin{aligned} \partial _tu - \partial _x^5u = u\partial _x^2u. \end{aligned}$$

(7.10)

The IVP associated to this equation is locally well-posed in $H^s(\mathbb {R}), s\ge 2$, using the contraction mapping principle. However, our modification to the proof of Theorem 1 will require the use of weighted Sobolev spaces.

After integrating by parts and supressing coefficients

$$\begin{aligned} B = \int u\left( \partial _x^{l+1}u\right) ^2\chi (x+\nu t) \; dx + \int \partial _x^2u\left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \tilde{B} \end{aligned}$$

(7.11)

where $\tilde{B}$ is controlled by induction. Combining with (7.8), after integrating (7.7) in time and using the hypothesis on the initial data we have

$$\begin{aligned} y(t):= & {} \int \left( \partial _x^lu\right) ^2\chi (x+\nu t) \; dx + \int _0^t\int \left( \partial _x^{l+2}u\right) ^2 \chi '(x+\nu \tau ) \; dxd\tau \nonumber \\\le & {} c_0 + \int _0^t\int \partial _x^2u\left( \partial _x^lu\right) ^2\chi (x+\nu \tau )\;dxd\tau + \left| \int _0^t\int u\left( \partial _x^{l+1}u\right) ^2\chi (x+\nu \tau )\;dxd\tau \right| \nonumber \\\le & {} c_0 + c_1\int _0^t y(\tau ) \; d\tau + \left| \int _0^t\int u\left( \partial _x^{l+1}u\right) ^2\chi (x+\nu \tau )\;dxd\tau \right| . \end{aligned}$$

(7.12)

Focusing on the last term in the above line,

$$\begin{aligned}&\left| \int _0^t\int u\left( \partial _x^{l+1}u\right) ^2\chi (x+\nu \tau )\;dxd\tau \right| \nonumber \\&\quad \le \left( \sum _{j\in \mathbb {Z}} \sup _{\begin{array}{c} 0 \le t \le T \\ j\le x \le j+1 \end{array}} |u(x,t)| \right) \left( \sup _{j\in \mathbb {Z}} \int _0^T\int _j^{j+1}\left( \partial _x^{l+1}u\right) ^2\chi (x+\nu \tau )\;dxd\tau \right) .\qquad \quad \end{aligned}$$

(7.13)

We check three cases to show the inductive case $l-1$ bounds the second factor. First, the integral vanishes for $j+1<\varepsilon -\nu T$. For $\varepsilon <j$ we apply the inductive hypothesis with $\nu =0$. Otherwise we utilize a pointwise bound on $\chi$

$$\begin{aligned} \int _0^T\int _j^{j+1}\left( \partial _x^{l+1}u\right) ^2\chi (x+\nu \tau )\;dxd\tau \lesssim \int _0^T\int \left( \partial _x^{l+1}u\right) ^2\chi '(x+\nu \tau ;\varepsilon /5,\nu T+ \varepsilon )\;dxd\tau . \end{aligned}$$

The technique for bounding the first factor is described in the next theorem. In general, there exists a nonnegative integer n depending on the form of the polynomial Q such that the following quantities must be estimated:

$$\begin{aligned} \sum _{j\in \mathbb {Z}} \sup _{\begin{array}{c} 0 \le t \le T \\ j\le x \le j+1 \end{array}} \left| \partial _x^ku(x,t)\right| , \quad k=0,1,\ldots ,n, \end{aligned}$$

assuming u is a Schwarz solution of IVP (7.1). With such an estimate in hand, the result follows by an application of Gronwall’s inequality. $\square$

Theorem 5

Let $k\in \mathbb {Z}^+\cup \{0\}$ and u be a Schwartz solution of the IVP (7.1) corresponding to initial data $u_0\in \mathscr {S}(\mathbb {R})$. Then there exists a nonnegative integer $m_0$ (depending on Q and k) and positive real number $s_0\ge 2m_0$ such that

$$\begin{aligned} \sum _{j\in \mathbb {Z}} \sup _{\begin{array}{c} 0 \le t \le T \\ j\le x \le j+1 \end{array}} \left| \partial _x^ku(x,t)\right| \le c\left( T;\Vert u_0\Vert _{X_{s_0,m_0}}\right) . \end{aligned}$$

The idea is to apply a Sobolev type inequality in the t-variable and show that the resulting summation converges by imposing enough spatial decay on the solution. Acheiving this goal requires the following lemma.

Lemma 6

If $f \in C^2(\mathbb {R}^2)$, then

$$\begin{aligned} \sup _{\begin{array}{c} 0 \le t \le T \\ 0 \le x \le L \end{array}} |f(x,t)|\le & {} \int _0^T\int _0^L |\partial _{xt} f(y,s)| \; dyds + \frac{1}{TL} \int _0^T\int _0^L |f(y,s)| \; dyds\\&\frac{1}{L} \int _0^T\int _0^L |\partial _tf(y,s)| \; dyds + \frac{1}{T} \int _0^T\int _0^L |\partial _xf(y,s)| \; dyds \end{aligned}$$

for any $L,T>0$.

We now turn to the proof of Theorem 5.

Proof

For concreteness, we show details for $k=0$. Applying Lemma 6,

$$\begin{aligned} \sum _{j\in \mathbb {Z}} \sup _{\begin{array}{c} 0 \le t \le T \\ j\le x \le j+1 \end{array}} |u(x,t)| \lesssim _T \Vert \partial _{xt}u\Vert _{L_T^1 L_x^1} + \Vert \partial _xu\Vert _{L_T^1 L_x^1} + \Vert \partial _tu\Vert _{L_T^1 L_x^1} + \Vert u\Vert _{L_T^1 L_x^1}. \end{aligned}$$

Focusing on the worst term $\Vert \partial _{xt}u\Vert _{L_T^1L_x^1}$ and applying

$$\begin{aligned} \Vert f\Vert _1 \le \Vert f\Vert _2 + \Vert xf\Vert _2 \end{aligned}$$

we arrive at

$$\begin{aligned} \Vert \partial _{xt}u\Vert _{L_T^1L_x^1} \lesssim _T \Vert \partial _{xt}u\Vert _{L_T^\infty L_x^2} + \Vert x\partial _{xt}u\Vert _{L_T^\infty L_x^2}. \end{aligned}$$

Looking at the second term and using the differential equation we have

$$\begin{aligned} \Vert x\partial _{xt}u\Vert _2 \le \Vert x\partial _x^6u(t)\Vert _2 + \Vert x\partial _x(u\partial _x^2u)\Vert _2 =: A+ B. \end{aligned}$$

Then

$$\begin{aligned} A^{2}= & {} \int x^{2}\left( \partial _x^6u\right) ^2dx\\= & {} \int u\partial _x^6\left( x^2\partial _x^6u\right) dx\\= & {} \int x^{2}u\partial _x^{12}udx +12\int xu\partial _x^{11}udx +30\int u\partial _x^{10}udx\\\lesssim & {} \Vert x^{2}u\Vert _2\left\| \partial _x^{12}u\right\| _2 + \Vert xu\Vert _2\left\| \partial _x^{11}u\right\| _2 + \Vert u\Vert _2\left\| \partial _x^{10}u\right\| _2. \end{aligned}$$

and so we impose $s_0\ge 12, m_0\ge 4$ (compared to the $H^2(\mathbb {R})$ local well-posedness). The estimates for the remaining terms are similar, completing the case $k=0$. $\square$