Semilinear p-Evolution Equations in Weighted Sobolev Spaces

Abstract

We consider the initial value problem for a class of semilinear p-evolution equations with (t, x)-depending coefficients. Under suitable decay conditions for |x|→∞ on the imaginary part of the coefficients, we prove local in time well posedness of the Cauchy problem in suitable weighted Sobolev spaces.

To Massimo Cicognani and Michael Reissig in occasion of their 60-th birthday

1 Introduction

In the present paper we deal with the semilinear Cauchy problem

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \begin{cases} P_u(D)u(t,x)=f(t,x),& (t,x)\in[0,T]\times\mathbb R\\ u(0,x)=u_0(x),&\displaystyle x\in\mathbb R \end{cases} \end{array} \end{aligned} $$

(1)

for the first order p-evolution operator

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} P_u(D)u=P(t,x,u(t,x),D_t,D_x)u:=& &\displaystyle D_tu +a_p(t)D_x^pu+\displaystyle\sum_{j=0}^{p-1}a_j(t,x,u) D_x^ju\quad \end{array} \end{aligned} $$

(2)

where $D=\frac 1i \partial$, $p \in \mathbb N, p \geq 2,$$a_p\in C([0,T], \mathbb R)$, a_j are for 0 ≤ j ≤ p − 1 continuous in time functions with values in $C^\infty (\mathbb R\times \mathbb C)$, and moreover the functions x → a_j(t, x, w) are in $\mathcal B^\infty (\mathbb R)$ (i.e. uniformly bounded together with all their derivatives).

For p = 2 our analysis will concern semilinear Schrödinger equations of the form

$$\displaystyle \begin{aligned}D_t u+ D_x^2 u + a_1(t,x,u)D_x u +a_0(t,x,u)= f(t,x).\end{aligned}$$

For p = 3, the most important model is represented by the Korteweg-de Vries equation describing the propagation of monodimensional waves of small amplitudes in waters of constant depth:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \partial_tu=\frac 32\sqrt{\frac gh}\partial_x\left(\frac 12u^2+\frac 23 \alpha u+\frac 13\sigma\partial_x^2u\right), \end{array} \end{aligned} $$

that can be written in the form (1) as

$$\displaystyle \begin{aligned}D_t u + \frac 12 \sqrt{\frac{g}{h}} \sigma D_{x}^3u - \sqrt{\frac{g}{h}} \left( \alpha+ \frac 32 u \right) D_x u = 0.\end{aligned}$$

Here u represents the wave elevation with respect to the water’s surface, g is the gravity constant, h the (constant) level of water, α a fixed small constant and $\sigma =\frac {h^3}{3}-\frac {Th}{\rho g}$, with T the surface tension, ρ the density of the fluid. Assuming the level of water h depending on x, we are led to an operator with space-depending coefficients that can be applied to study the evolution of the wave when the depth of the seabed is variable, cf. [1].

Since a_p is real valued, the principal symbol (in the sense of Petrowski) of P, given by τ + a_p(t)ξ^p, has the real characteristic root τ = −a_p(t)ξ^p; by the Lax-Mizohata theorem, real characteristics are necessary for the existence of a unique solution in Sobolev spaces of the Cauchy problem (1) in a neighborhood of t = 0, for any p ≥ 1. Moreover, whenever the lower order coefficients $a_j(t,x,w)\in \mathbb C$ for 0 ≤ j ≤ p − 1, decay conditions as |x|→∞ are necessary on the a_j for well-posedness in Sobolev spaces, see [6, 15] respectively for p = 2, p arbitrary.

Well-posedness for the Cauchy problem (1), (2) in $H^\infty (\mathbb R)=\cap _s H^s(\mathbb R)$, where $H^s(\mathbb R)$ is the usual Sobolev space on L², has been proved in the paper [1] under suitable decay conditions at infinity for the a_j, 0 ≤ j ≤ p − 1, relying on the linear results of [5]; in this paper, despite very precise decay assumptions on the coefficients, the authors have no information at all about the behavior at infinity of the solution.

In the last years, we started to study linear p-evolution equations in weighted Sobolev spaces, see [3, 4] and to state a relation between the behavior at infinity of the data and the one of the solution. Here we are interested to extend part of these results to the semilinear case, that is to give decay conditions on the coefficients of P_u(D) that are sufficient for the local in time well-posedness of the Cauchy problem (1) in suitable weighted Sobolev spaces.

Namely, fixed $s_1,s_2\in \mathbb R,$ we define $H^{s_1,s_2}(\mathbb R)$ as the space of all $u \in \mathscr {S}'(\mathbb R)$ such that $\|u \|{ }_{s_1,s_2}:=\|\langle x \rangle ^{s_{2}}\langle D \rangle ^{s_{1 }}u\|{ }_{L^{2}} <\infty$ where we denote by $\langle D \rangle ^{s_1}$ the Fourier multiplier with symbol $\langle \xi \rangle ^{s_1}:=(1+\xi ^2)^{s_1/2}$. This space is a Hilbert space endowed with the inner product

$$\displaystyle \begin{aligned}\langle u,v \rangle_{s_1,s_2}:= \langle \langle x \rangle^{s_{2}}\langle D \rangle^{s_{1 }}u, \langle x \rangle^{s_{2}}\langle D \rangle^{s_{1 }}v \rangle_{L^2}\end{aligned}$$

which induces the norm $\| \cdot \|{ }_{s_1,s_2}$. We have $H^{0,0}(\mathbb R)=L^2(\mathbb R)$ and we shall denote the L² norm simply by ∥⋅∥. An equivalent norm on $H^{s_1,s_2}(\mathbb R)$ is given by $|||u |||{ }_{s_1,s_2}:= \|\langle D \rangle ^{s_1} \langle x \rangle ^{s_2}u\|{ }_{L^2}.$ Notice that for s₂ = 0 we recapture the standard Sobolev spaces and that the obvious inclusions $H^{s_1,s_2}(\mathbb R)\subseteq H^{t_1,t_2}(\mathbb R)$ for every s₁ ≥ t₁, s₂ ≥ t₂ hold. We also recall that $H^{s_1,s_2}(\mathbb R)$ is an algebra with respect to multiplication for s₁ > 1∕2 and s₂ ≥ 0, cf. [2, Proposition 2.2]. For every given $s_1\in \mathbb R$ (resp. $s_2\in \mathbb R$) we define

$$\displaystyle \begin{aligned}H^{s_1,\infty}(\mathbb R):=\bigcap_{s_2 \in \mathbb R}H^{s_1,s_2}(\mathbb R),\quad \mathrm{resp.}\quad H^{\infty,s_2}(\mathbb R):=\bigcap_{s_1 \in \mathbb R}H^{s_1,s_2}(\mathbb R).\end{aligned}$$

We remark that $H^{s_1,\infty }(\mathbb R)$ consists of functions with the same decay as the functions of $\mathscr S(\mathbb R)$ but with a limited regularity, while $H^{\infty ,s_2}(\mathbb R)$ consists of functions in $H^\infty (\mathbb R)$ with a prescribed decay as |x|→∞. As it will be shown in Sect. 2, these two spaces are graded Fréchet spaces endowed with the increasing families of seminorms

$$\displaystyle \begin{aligned}|u|{}_{s_1,k}:=\max_{s_2\leq k}\|u\|{}_{s_1,s_2},\quad \mathrm{resp.}\quad |u|{}_{k,s_2}:=\max_{s_1\leq k}\|u\|{}_{s_1,s_2},\quad k\in\mathbb N,\end{aligned}$$

and they are tame (see Definition 1). Finally, we notice that

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \bigcap_{s_1 \in \mathbb R}H^{s_1,\infty}(\mathbb R)= \bigcap_{s_2 \in \mathbb R}H^{\infty,s_2}(\mathbb R)= \mathscr{S}(\mathbb R). \end{array} \end{aligned} $$

(3)

The main result of the paper is the following.

Theorem 1

Let P(t, x, D_t, D_x) be an operator of the form (2). Assume that there exist a constant C > 0 and a function$\gamma :\ \mathbb C\to \mathbb R^+$of class C⁷such that for all$(t,x,w)\in [0,T]\times \mathbb R\times \mathbb C$, $\beta ,\delta \in \mathbb N$the following conditions hold:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & &\displaystyle \quad a_p(t) \quad \mathit{\mbox{is real valued and}} \quad a_p(t)\neq 0, \qquad t \in [0,T]; \end{array} \end{aligned} $$

(4)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & &\displaystyle \quad |\partial_w^\delta\partial_x^\beta\mathop{\mathrm{Im}}\nolimits a_j(t,x,w)|\leq C \gamma(w)\langle x\rangle^{-\frac j{p-1}-|\beta|},\quad 0\leq j\leq p-1; \end{array} \end{aligned} $$

(5)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & &\displaystyle \quad |\partial_w^\delta\partial_x^\beta\mathop{\mathrm{Re}}\nolimits a_j(t,x,w)|\leq C \gamma(w)\langle x\rangle^{-|\beta|},\quad 0\leq j\leq p-1. \end{array} \end{aligned} $$

(6)

Then, for every given s ₂ ≥ 3p − 2, the Cauchy problem (1) is well-posed locally in time in$H^{\infty ,s_2}(\mathbb R)$: namely for all$f\in C([0,T];H^{\infty ,s_2}(\mathbb R))$and$u_0\in H^{\infty ,s_2}(\mathbb R)$, there exists 0 < T^∗≤ T and a unique solution$u\in C^1([0,T^*]; H^{\infty ,s_2}(\mathbb R))$of (1).

Remark 1

With respect to [1], in Theorem 1 from the decay at infinity of the data we can estimate the decay of the solution as |x|→∞. Indeed, by [1] we know that if the data are in H^∞ (and the decay conditions are satisfied), then the solution belongs to H^∞, too; Theorem 1 states that if the data are in $H^{\infty ,s_2}$ for s₂ large enough, then also $u\in H^{\infty ,s_2}$.

The idea of the proof of Theorem 1 is the following: to show the existence of a unique solution to the semilinear equation (1) in $H^{\infty ,s_2}$, we first linearize it, fixing a function $u\in C([0,T], H^{\infty ,s_2}(\mathbb R))$ with $s_2\in \mathbb R$ large enough, then we solve the linear Cauchy problem in the unknown v(t, x)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \begin{cases} P_u(D)v(t,x)=f(t,x),& (t,x)\in[0,T]\times\mathbb R\\ v(0,x)=u_0(x),&\displaystyle x\in\mathbb R \end{cases} \end{array} \end{aligned} $$

(7)

in $H^{\infty ,s_2}(\mathbb R)$; finally, inspired by [6], [10] and [12], we apply the Nash-Moser theorem to obtain the existence of a unique solution of (1) in the tame space $H^{\infty ,s_2}(\mathbb R)$. We remark that we cannot apply to the Cauchy problem (1), (2) a usual fixed point scheme in Banach spaces since the linearized problem (7) has a unique solution which presents a loss of regularity and/or a different behavior at infinity with respect to the data. Thus the problem (7) is not well posed in $H^{s_1,s_2}$; however it turns out to be well posed in $H^{\infty ,s_2}(\mathbb R)$ which is a tame Fréchet space, so there we can apply the Nash Moser theorem.

Remark 2

In the linear case treated in [3], as a consequence of the energy estimates in weighted Sobolev spaces, we also obtained that the Cauchy problem is well posed in $\mathscr {S}(\mathbb R)$ and $\mathscr {S}'(\mathbb R)$. In the semilinear case, we are not able to prove in the same way well posedness in $\mathscr {S}(\mathbb R)$. In fact, if the data of the problem are Schwartz functions, they belong in particular to $H^{\infty , s_2}(\mathbb R)$ for every s₂ > 0, however, in the semilinear case, the upper bound T^∗ of the interval of existence of the solution may depend on s₂ and possibly tends to 0 when s₂ → +∞.

Remark 3

The techniques used in this paper may be adapted to study semilinear p-evolution equations in higher space dimension x at least in some particular cases as, for instance, Schrödinger-type equations (p = 2). For this type of equations, at least the linear theory is well established in general space dimension, cf. [8, 9, 16] and it could be easily applied to the analysis of the linearized Cauchy problem (7). We will treat this problem for general p-evolution equations in a future paper.

2 Preliminaries: SG-Calculus and Nash Moser Theorem

2.1 SG-Calculus

We recall here the definition and the main properties of the SG classes of pseudodifferential operators. In view of the purposes of this paper we shall state them for symbols defined on $\mathbb R^2$, but they have obvious extension in higher dimension. For this generalization and for more details on these classes we refer to [11, 19, 20]. Fixed $m_1,m_2 \in \mathbb R$, the space $\mathbf {SG}^{m_{1},m_{2}}(\mathbb R^{2})$ is the space of all functions $p(x,\xi ) \in C^{\infty }(\mathbb R^{2})$ satisfying the following estimates:

$$\displaystyle \begin{aligned} \sup_{(x,\xi) \in \mathbb R^{2}}\langle \xi \rangle^{-m_{1}+\alpha} \langle x \rangle^{-m_{2}+\beta}|\partial_{\xi}^{\alpha}\partial_{x}^{\beta} p(x,\xi)| <\infty \end{aligned} $$

(8)

for every $\alpha , \beta \in \mathbb N.$ We can associate to every $p \in \mathbf {SG}^{m_1,m_2}(\mathbb R^{2})$ the pseudodifferential operator defined by

$$\displaystyle \begin{aligned} Pu(x)=p(x,D)u(x) = (2\pi)^{-d} \int_{\mathbb{R}^d} e^{i\langle x,\xi\rangle} p(x,\xi) \hat{u}(\xi)\, d\xi. \end{aligned} $$

(9)

If $p \in \mathbf {SG}^{m_{1},m_{2}}(\mathbb R^{2})$, then the operator p(x, D) is a linear continuous map from $\mathscr {S}(\mathbb R)$ to $\mathscr {S}(\mathbb R)$ and extends to a linear continuous map from $\mathscr {S}'(\mathbb R)$ to $\mathscr {S}'(\mathbb R)$ and from $H^{s_{1},s_{2}}(\mathbb R)$ to $H^{s_{1}-m_{1},s_{2}-m_{2}}(\mathbb R)$ for every $s_1,s_2 \in \mathbb R$. We also recall the following result concerning the composition and the adjoint of SG operators.

Proposition 1

Let$p \in \boldsymbol {SG}^{m_{1},m_{2}}(\mathbb R^{2})$and$q \in \boldsymbol {SG}^{m^{\prime }_{1},m^{\prime }_{2}}(\mathbb R^{2})$. Then there exists a symbol$s \in \boldsymbol {SG}^{m_{1}+m^{\prime }_{1},m_{2}+m^{\prime }_{2}}(\mathbb R^{2})$such that p(x, D)q(x, D) = s(x, D) + R where R is a smoothing operator$\mathscr {S}'(\mathbb R) \to \mathscr {S}(\mathbb R).$Moreover, s has the following asymptotic expansion

$$\displaystyle \begin{aligned}s(x,\xi) \sim \sum_{\alpha} \alpha!^{-1}\partial_{\xi}^{\alpha}p(x,\xi) D_{x}^{\alpha}q(x,\xi)\end{aligned}$$

i.e. for every N ≥ 1, we have

$$\displaystyle \begin{aligned}s(x,\xi) - \sum_{|\alpha|<N} \alpha!^{-1}\partial_{\xi}^{\alpha}p(x,\xi) D_{x}^{\alpha}q(x,\xi) \in \boldsymbol{SG}^{m_{1}+m^{\prime}_{1}-N,m_{2}+m^{\prime}_{2}-N}(\mathbb R^{2}).\end{aligned}$$

Proposition 2

Let$p \in \boldsymbol {SG}^{m_{1},m_{2}}(\mathbb R^{2})$and let P^∗be the L²-adjoint of p(x, D). Then there exists a symbol$p^{\ast } \in \boldsymbol {SG}^{m_{1},m_{2}}(\mathbb R^{2})$such that P^∗ = p^∗(x, D) + R′, where R′ is a smoothing operator$\mathscr {S}'(\mathbb R) \to \mathscr {S}(\mathbb R).$Moreover, p^∗has the following asymptotic expansion

$$\displaystyle \begin{aligned}p^{\ast}(x,\xi) \sim \sum_{\alpha} \alpha!^{-1}\partial_{\xi}^{\alpha}D_{x}^{\alpha}\overline{p(x,\xi)}\end{aligned}$$

i.e. for every N ≥ 1, we have

$$\displaystyle \begin{aligned}p^{\ast}(x,\xi) - \sum_{|\alpha|<N} \alpha!^{-1}\partial_{\xi}^{\alpha}D_{x}^{\alpha}\overline{p(x,\xi)} \in \boldsymbol{SG}^{m_{1}-N,m_{2}-N}(\mathbb R^{2}).\end{aligned}$$

We will denote in the sequel by $S^m(\mathbb R^{2}), m \in \mathbb R,$ the class of symbols $p(x,\xi ) \in C^{\infty }(\mathbb R^{2})$ satisfying

$$\displaystyle \begin{aligned} \sup_{(x,\xi) \in \mathbb R^{2}}\langle \xi \rangle^{-m+\alpha}|\partial_{\xi}^{\alpha}\partial_{x}^{\beta} p(x,\xi)| <\infty, \end{aligned}$$

for every $\alpha , \beta \in \mathbb N.$ We observe that the following inclusion holds

$$\displaystyle \begin{aligned} \mathbf{SG}^{m_1,m_2}(\mathbb R^{2}) \subset S^{m_1}(\mathbb R^{2}) \end{aligned} $$

(10)

for every $m_1 \in \mathbb R, m_2 \leq 0$.

The following theorem has been proved in [3, Theorem 2.3], and provides an extension to pseudodifferential operators of SG-type of the well known sharp Gårding theorem.

Theorem 2

Let m ₁ ≥ 0, m₂ ≤ 0, $a \in \boldsymbol {SG}^{m_1,m_2}(\mathbb R^{2})$such that$\mathop {\mathrm {Re}} \nolimits a(x,\xi )\geq 0$if |ξ|≥ C for some positive C. Then there exist pseudo-differential operators Q = q(x, D), R = r(x, D) and R₀ = r₀(x, D) with symbols, respectively,$q \in \boldsymbol {SG}^{m_1,m_2}(\mathbb R^{2})$, $r \in \boldsymbol {SG}^{m_1-1,m_2}(\mathbb R^{2})$and$r_0 \in S^0(\mathbb R^{2})$such that

$$\displaystyle \begin{aligned} a(x,D)=q(x,D)+r(x,D)+r_0(x,D), {} \end{aligned} $$

(11)

$$\displaystyle \begin{aligned} \mathop{\mathrm{Re}}\nolimits\langle q(x,D) u,u\rangle\geq0\qquad \forall u\in \mathscr{S}(\mathbb R) \end{aligned} $$

(12)

and

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} r(x,\xi)= \psi_1(\xi)D_x a(x,\xi)+ \sum_{2\leq \alpha+\beta\leq 2m_1-1}\psi_{\alpha,\beta}(\xi) \partial_\xi^\alpha D_x^\beta a(x,\xi) \end{array} \end{aligned} $$

(13)

for some real valued functions ψ ₁, ψ_α,βwith$\psi _1 \in \boldsymbol {SG}^{-1,0}(\mathbb R^{2})$and$\psi _{\alpha , \beta } \in \boldsymbol {SG}^{\alpha -\beta /2,0}(\mathbb R^{2})$depending only on ξ.

We remark that the terms in (13) can be re-arranged so that we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} r(x,\xi)&=&\sum_{j=1}^{m-1}r_{j}(x,\xi), \end{array} \end{aligned} $$

(14)

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} r_{j}(x,\xi)&=&\left\{ \begin{array}{ll} \psi_1(\xi)D_xa(x,\xi)+\!\!\!\displaystyle\sum_{2\leq \alpha+\beta\leq3}\psi_{\alpha,\beta}(\xi)\partial_\xi^\alpha D_x^\beta a(x,\xi), & \quad \!\!\!j=m-1, \\ \displaystyle\sum_{2(m-j)\leq \alpha+\beta\leq2(m-j)+1}\!\!\!\psi_{\alpha,\beta}(\xi)\partial_\xi^\alpha D_x^\beta a(x,\xi), & \quad \!\!\!1\leq j\leq m-2. \end{array} \right. \end{array} \end{aligned} $$

(15)

We also remark that Theorem 2 implies the well-known sharp Gårding inequality

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \mathop{\mathrm{Re}}\nolimits\langle A(x,D_x)u,u\rangle\geq-c\|u\|{}_{(m-1)/2,0}^2 \end{array} \end{aligned} $$

(16)

for some fixed constant c > 0 (cf. [17, Theorem 4.4]).

We recall here also the Fefferman-Phong inequality (cf. [13]):

Theorem 3

Let$A(x,\xi )\in S^{m}(\mathbb R^2)$with A(x, ξ) ≥ 0. Then

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \mathop{\mathrm{Re}}\nolimits\langle A(x,D_x)u,u\rangle\geq-c\|u\|{}_{(m-2)/2,0}^2\qquad \forall u\in H^{m,0} \end{array} \end{aligned} $$

(17)

for some c > 0.

We remark that, by Lerner and Morimoto [18], for m = 2 the constant c in (17) depends only on $\max\limits _{|\alpha |+|\beta |\leq 7}C_{\alpha ,\beta }$ for $C_{\alpha ,\beta }:=\displaystyle\sup _{x,\xi \in \mathbb R}|\partial _\xi ^\alpha \partial _x^\beta A(x,\xi )|\langle \xi \rangle ^{-2+\alpha }$.

2.2 Tame Fréchet Spaces and the Nash Moser Theorem

We recall here the notions of tame space, tame maps, and the statement of the Nash-Moser inversion theorem, see [14] for further details. Moreover, we show that, for every fixed $s_1,s_2\in \mathbb R$, $H^{s_1,\infty }$ and $H^{\infty ,s_2}$ are tame spaces.

A graded Fréchet space X is a Fréchet space endowed with a grading, i.e. an increasing sequence of semi-norms:

$$\displaystyle \begin{aligned} \begin{array}{rcl} |x|{}_n\leq|x|{}_{n+1},\qquad \forall n\in\mathbb N_0, \, x\in X. \end{array} \end{aligned} $$

Example 1

Given a Banach space B, consider the space Σ(B) of all sequences $\{v_k\}_{k\in \mathbb N_0}\subset B$ such that

$$\displaystyle \begin{aligned} \begin{array}{rcl} |\{v_k\}|{}_n:=\sum_{k=0}^{+\infty}e^{nk}\|v_k\|{}_B<+\infty\qquad \forall n\in\mathbb N_0. \end{array} \end{aligned} $$

We have that Σ(B) is a graded Fréchet space with the topology induced by the family of seminorms |⋅|_n (which is in fact a grading on Σ(B)).

We say that a linear map L : X → Y between two graded Fréchet spaces is a tame linear map if there exist $r,n_0\in \mathbb N$ such that for every integer n ≥ n₀ there exists a constant C_n > 0, depending only on n, s.t.

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} |Lx|{}_n\leq C_n|x|{}_{n+r}\qquad \forall x\in X. \end{array} \end{aligned} $$

(18)

The numbers n₀ and r are called respectively base and degree of the tame estimate (18).

Definition 1

A graded Fréchet space X is said to be tame if there exist a Banach space B and two tame linear maps L₁ : X → Σ(B) and L₂ :  Σ(B) → X such that L₂ ∘ L₁ is the identity on X.

Obviously, given a graded Fréchet space X and a tame space Y , if there exist two linear tame maps L₁ : X → Y and L₂ : Y → X such that L₂ ∘ L₁ is the identity on X, then also X is a tame space.

Lemma 1

The spaces $H^{s_1,\infty }$ and $H^{\infty ,s_2}$ are tame.

Proof

We first recall that $H^\infty :=\bigcap _{s\in \mathbb R}H^s$ endowed with the seminorms |f|_n :=max_s≤n∥f∥_s for every $n\in \mathbb N$ is a tame Fréchet space, cf. [10]. Moreover the map $L:H^\infty \to \, H^{\infty ,s_2}$ defined by $L(f)=\langle x \rangle ^{-s_2}f$ is a tame isomorphism since for every n = 0, 1, 2, … we have:

$$\displaystyle \begin{aligned} \begin{array}{rcl} |L(f)|{}_{n, s_2}& =&\displaystyle \max_{s_1\leq n}\|L(f)\|{}_{s_1,s_2}=\max_{s_1\leq n}\|\langle x \rangle^{-s_2}f\|{}_{s_1,s_2} \\ & \leq &\displaystyle C_n \max_{s_1\leq n}|||\langle x \rangle^{-s_2}f |||{}_{s_1,s_2}=|f|{}_n \end{array} \end{aligned} $$

and

$$\displaystyle \begin{aligned}|f|{}_n=\max_{s_1\leq n}\|f\|{}_{s_1}\leq C^{\prime}_n \max_{s_1\leq n}\|\langle x \rangle^{-s_2}f\|{}_{s_1,s_2}=|L(f)|{}_{n,s_2}.\end{aligned}$$

Thus, $H^{\infty ,s_2}$ is a tame space. $H^{s_1,\infty }$ is also tame, since the Fourier transform $\mathcal F$ is an isomorphism between $H^{s_1,s_2}$ and $H^{s_2,s_1}$, and $\|\mathcal F(f)\|{ }_{s_2,s_1}=\|f\|{ }_{s_1,s_2}$; by this, it is easy to prove that $\mathcal F: H^{s_1,\infty }\to \,H^{\infty ,s_2}$ defines a tame map with tame inverse given by the inverse Fourier transform. □

Given now a nonlinear map T : U → Y where U ⊂ X and X, Y are graded spaces, we say that T satisfies a tame estimate of degree r and base n₀ if for every integer n ≥ n₀ there exists a constant C_n > 0 such that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} |T(u)|{}_n\leq C_n(1+|u|{}_{n+r})\qquad \forall u\in U. \end{array} \end{aligned} $$

(19)

We say that T is tame if it satisfies a tame estimate (19) in a neighborhood of each point u ∈ U (with constants r, n₀ and C_n which may depend on the neighbourhood).

Notice that a linear map is tame if and only if it is a tame linear map. Given a map T : U ⊂ X → Y , we define the Fréchet derivative DT(u)v of T at u ∈ U in the direction v ∈ X by

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} DT(u)v:=\lim_{\epsilon\to0}\frac{T(u+\epsilon v)-T(u)}{\epsilon}, \end{array} \end{aligned} $$

(20)

and we say that T is C¹(U) if the limit (20) exists and the derivative DT : U × X → Y is continuous. We can also define recursively the higher order Fréchet derivatives DⁿT : U × Xⁿ → Y of T, cf. [14]; we say that T is C^∞(U) if all the Fréchet derivatives of T exist and are continuous. A smooth tame map T : U → Y defined on an open subset U of X is a C^∞ map such that DⁿT is tame for all $n\in \mathbb N_0$.

It is known that sums and compositions of smooth tame maps are smooth tame, and, moreover, linear and nonlinear partial differential operators and integration are smooth tame maps, see [14] for the proofs of these results. Finally we recall the statement of Nash-Moser inversion theorem in the tame Fréchet spaces category, which will be used in the sequel to approach the Cauchy problem (1).

Theorem 4 (Nash-Moser-Hamilton)

Let X, Y be tame spaces, U an open subset of X and T : U → Y a smooth tame map. If the equation DT(u)v = h has a unique solution v := S(u, h) for all u ∈ U and h ∈ Y , and if S : U × Y → X is smooth tame, then T is locally invertible and each local inverse is smooth tame.

3 Well Posedness for the Linearized Cauchy Problem

The following theorem is the key to prove the main result of this paper. It deals with the linear Cauchy problem (7), and proves that if the data of (7) are chosen in the Sobolev space $H^{s_1,s_2}$, $s_1,s_2\in \mathbb R,$ then there exists a unique solution $v(t)\in H^{s_1-2\delta \eta (p-1),s_2-2\delta \epsilon }$ for some δ > 0 and for every 0 ≤ 𝜖, η ≤ 1 such that 𝜖 + η = 1.

Theorem 5

Under the assumptions of Theorem1, there exists δ > 0 such that for every$u\in C([0,T];H^{3p-1, 3p-2}(\mathbb R))$, $f\in C([0,T]; H^{s_1,s_2}(\mathbb R))$and$u_0\in H^{s_1,s_2}(\mathbb R)$, there exists a unique solution v of (7) such that$v\in C^1([0,T]; H^{s_1-2\delta \eta (p-1),s_2-2\delta \epsilon }(\mathbb R))$for every 𝜖, η ∈ [0, 1] with 𝜖 + η = 1. Moreover v satisfies the following energy estimate:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & &\displaystyle \|v(t,\cdot)\|{}^2_{s_1-2\delta\eta(p-1),s_2-2\delta\epsilon} \\ & &\displaystyle \hskip+1cm\leq C_{s_1,s_2,\gamma}e^{(1+\|u\|{}^{3p-2}_{3p-1, 3p-2})t}\left(\!\|u_0\|{}^2_{s_1,s_2}+ \int_0^t\|f(\tau,\cdot)\|{}^2_{s_1,s_2} \,d\tau\!\right)\, \forall t\in[0,T]. \end{array} \end{aligned} $$

(21)

Remark 4

Notice that the solution v presents the loss 2δη(p − 1) in the first Sobolev index and the loss 2δ𝜖 in the second one. In the case s₂ = 0, 𝜖 = 0, η = 1 we recapture the result of [1, Theorem 2.1]. Moreover, in the linear case (i.e., if (7) does not depend on u), we can obtain either well-posedness with loss of 2δ(p − 1) derivatives and no loss of decay (take η = 1 and 𝜖 = 0), or the result of [3], that is well-posedness without loss of derivatives but with loss of decay 2δ (take η = 0 and 𝜖 = 1). We can also obtain all the intermediate estimates. A similar result has been proved in [7], where intermediate estimates for Schrödinger equations (p = 2) have been proved in Gevrey classes.

The proof of Theorem 5 consists in choosing an appropriate and invertible change of variable

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} v(t,x)=e^\Lambda(x,D)w(t,x) \end{array} \end{aligned} $$

(22)

which transforms the Cauchy problem (7) into an equivalent Cauchy problem

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \begin{cases} P_\Lambda(t,x,u(t,x), D_t,D_x) w(t,x)=f_\Lambda(t,x) \quad (t,x)\in [0,T]\times\mathbb R\\ w(0,x)=u_{0,\Lambda}(x) \quad x\in\mathbb R \end{cases} \end{array} \end{aligned} $$

(23)

for

$$\displaystyle \begin{aligned}P_\Lambda:=(e^\Lambda)^{-1}Pe^\Lambda,\quad f_\Lambda:=(e^\Lambda)^{-1}f,\quad u_{0,\Lambda}:=(e^\Lambda)^{-1}u_0\end{aligned}$$

which is well-posed in L² (and therefore in all the weighted Sobolev spaces $H^{s_1,s_2}$). By the energy estimate in $H^{s_1,s_2}$ for the solution w to the Cauchy problem (23), we then deduce the energy estimate (21) from (22).

The operator Λ will be of the form

$$\displaystyle \begin{aligned}\Lambda (x,D)=\lambda_1(x,D)+\ldots+\lambda_{p-1}(x,D),\end{aligned}$$

so

$$\displaystyle \begin{aligned}P_\Lambda:=(e^{\lambda_1})^{-1}\cdots(e^{\lambda_{p-1}})^{-1}Pe^{\lambda_{p-1}}\cdots(e^{\lambda_1}),\end{aligned}$$

$$\displaystyle \begin{aligned}f_\Lambda:=(e^{\lambda_1})^{-1}\cdots(e^{\lambda_{p-1}})^{-1}f,\quad u_{0,\Lambda}:=(e^{\lambda_1})^{-1}\cdots(e^{\lambda_{p-1}})^{-1}u_0.\end{aligned}$$

We construct here below the transformation Λ and we point out its main properties in Proposition 3. Then we prove the invertibility of e^Λ in Proposition 4. In the subsequent Lemma 2 we show how to obtain the energy estimate (21) for the Cauchy problem (7) from the $H^{s_1,s_2}$ energy estimate for the Cauchy problem (23). After that, in Lemma 5 we state the regularity with respect to x, u of the coefficients a_j(t, x, u) of the linear operator (7), for 0 ≤ j ≤ p − 1. This section ends with the proof of Theorem 5.

Definition 2

For every k = 1, …, p − 1 we define the symbols

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \lambda_{p-k}(x,\xi):=M_{p-k}\omega\left(\frac\xi h\right)\langle\xi\rangle_h^{-k+1} \int_0^x \langle y\rangle^{-\frac{p-k}{p-1}}\psi\left( \frac{\langle y\rangle}{\langle\xi\rangle_h^{p-1}}\right)dy , \end{array} \end{aligned} $$

(24)

where h and M_p−k are positive constants such that h ≥ 1, $\omega \in C^\infty (\mathbb R)$ is such that

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} & &\displaystyle \omega(y)= \begin{cases} 0 &\displaystyle |y|\leq 1\\ |y|{}^{p-1}/y^{p-1}& |y|\geq 2 \end{cases}, \end{array} \end{aligned} $$

(25)

and $\psi \in C^\infty _0(\mathbb R)$ is such that 0 ≤ ψ(y) ≤ 1 for all $y\in \mathbb R$, ψ(y) = 1 for $|y|\leq \frac 12$, and ψ(y) = 0 for |y|≥ 1.

Proposition 3

There exists a constant C > 0 such that for every$(x,\xi ) \in \mathbb R^{2}$the following conditions hold:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} |\lambda_{p-1}(x,\xi)|& \leq&\displaystyle M_{p-1}\left(\log 2+\epsilon\log\langle x \rangle+\eta(p-1)\log\langle \xi \rangle_h\right) \\ & &\displaystyle \forall\epsilon,\eta \in [0,1] \; \epsilon+\eta=1; \end{array} \end{aligned} $$

(26)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} |\lambda_{p-k}(x,\xi)|& \leq&\displaystyle C M_{p-k}, \quad 2\leq k\leq p-1. \end{array} \end{aligned} $$

(27)

Moreover, for every α, β with (α, β) ≠ (0, 0), there exists C_α,β > 0 such that for |ξ| > 2h:

$$\displaystyle \begin{aligned} |\partial_\xi^\alpha\partial_x^\beta\lambda_{p-k}(x,\xi)|\leq C_{\alpha,\beta}\langle x\rangle^{-\beta} \langle\xi\rangle_h^{-\alpha}, \quad 1\leq k\leq p-1. \end{aligned} $$

(28)

Proof

We only prove (26) and (27); the inequality (28) can be deduced as in the proof of [5, Lemma 2.1]. Let $E=\{(y, \xi ) \in \mathbb R^2: \langle y\rangle \leq \langle \xi \rangle _h^{p-1}\}$. If x ∈ E, x > 0, then by (24), integrating we have:

$$\displaystyle \begin{aligned} \begin{array}{rcl} |\lambda_{p-1}(x,\xi)|& \leq &\displaystyle M_{p-1}\int_{0}^{x} \frac{1}{\sqrt{1+y^2}}dy \leq M_{p-1}\log\left(2\langle x \rangle\right) \\ & \leq&\displaystyle M_{p-1}(\ln 2+\log\langle x \rangle) \\ & \leq&\displaystyle M_{p-1}(\ln 2+\log\langle x \rangle^\epsilon\langle \xi \rangle_h^{\eta(p-1)}) \\ & \leq &\displaystyle M_{p-1}(\ln 2+\epsilon\log\langle x \rangle+\eta(p-1)\log\langle \xi \rangle_h) \end{array} \end{aligned} $$

for every 0 ≤ 𝜖, η ≤ 1, 𝜖 + η = 1. Analogously, if x∉E we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} |\lambda_{p-1}(x,\xi)|& \leq &\displaystyle M_{p-1}\int_{0}^{\sqrt{\langle \xi \rangle_h^{2(p-1)}-1}} \frac{1}{\sqrt{1+y^2}}dy \\ & \leq&\displaystyle M_{p-1}\ln (2\langle \xi \rangle_h^{p-1}) \\ & \leq &\displaystyle M_{p-1}(\ln 2+\log\langle x \rangle^\epsilon\langle \xi \rangle_h^{\eta(p-1)}) \\ & \leq&\displaystyle M_{p-1}(\ln 2+\epsilon\log\langle x \rangle+\eta(p-1)\log\langle \xi \rangle_h), \end{array} \end{aligned} $$

using the fact that for x∉E we have $\langle \xi \rangle _h^{p-1} < \langle x \rangle .$ Similar estimates can be obtained for x < 0. The estimate (27) can be proved by a similar argument. □

From Proposition 3 we obtain in particular that $e^{\pm \lambda _{p-1}} \in \mathbf {SG}^{ M_{p-1}\eta (p-1), M_{p-1}\epsilon }$ for every 𝜖, η ≥ 0 such that 𝜖 + η = 1 whereas for k = 2, …, p − 1, we have $e^{\pm \lambda _{p-k}} \in \mathbf {SG}^{0,0}(\mathbb R^2) \subset S^0(\mathbb R^2).$

Proposition 4

For every k = 1, …, p − 1, let λ_p−kbe defined by (24). There exists h_k ≥ 1 such that for every h ≥ h_kthe operator$e^{\lambda _{p-k}}(x,D)$is invertible and

$$\displaystyle \begin{aligned} \begin{array}{rcl} \left(e^{\lambda_{p-k}}(x,D)\right)^{-1}=e^{-\lambda_{p-k}}(x,D)(I+r_{p-k}(x,D)), \end{array} \end{aligned} $$

(29)

where I stands for the identity operator and r _p−k(x, D) is a pseudodifferential operator with principal symbol

$$\displaystyle \begin{aligned} \begin{array}{rcl}{}r_{p-k,-k}(x,\xi)=\partial_\xi\lambda_{p-k}(x,\xi)D_x\lambda_{p-k}(x,\xi)\in \boldsymbol{SG}^{-k,-\frac{p-k}{p-1}}.\end{array} \end{aligned} $$

(30)

Proof

We first observe that

$$\displaystyle \begin{aligned}e^{\lambda_{p-k}}(x,D)e^{-\lambda_{p-k}}(x,D)=I-\tilde r_{p-k}(x,D),\end{aligned}$$

where $\tilde r_{p-k}$ has principal symbol r_p−k,−k in (30). From (28) we have

$$\displaystyle \begin{aligned}|r_{p-k,-k}(x,\xi)|\leq C_kM_{p-k}^2h^{-1},\end{aligned}$$

and we similarly estimate the derivatives. We see that for h large enough, say h ≥ h_k, the operator $I-\tilde r_{p-k}$ is invertible on L² with inverse given by the Neumann series

$$\displaystyle \begin{aligned}\sum_{j\geq 0} \tilde r^j_{p-k}=I+r_{p-k},\end{aligned}$$

and the operator r_p−k has principal part (30). Thus,

$$\displaystyle \begin{aligned}e^{\lambda_{p-k}}(x,D)e^{-\lambda_{p-k}}(x,D)(I+r_{p-k})=I,\end{aligned}$$

and $e^{-\lambda _{p-k}}(x,D)(I+r_{p-k})$ is a right inverse of $e^{\lambda _{p-k}}(x,D)$. Similarly we can obtain that it is also a left inverse. □

Lemma 2

If the Cauchy problem (23) is$H^{s_1,s_2}$well posed, and the energy estimate

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \|w\|{}^2_{s_1,s_2}\leq C e^{(1+\|u\|{}_{3p-1,3p-2}^{3p-2})t}\left(\|u_{0,\Lambda}\|{}_{s_1,s_2}^2+\int_0^t \|f_\Lambda(\tau)\|{}_{s_1,s_2}^2d\tau\right) \end{array} \end{aligned} $$

(31)

holds for every t ∈ [0, T], then the Cauchy problem (7) admits a unique solution

$$\displaystyle \begin{aligned}v\in C([0,T]; H^{s_1-2\delta\eta(p-1), s_2-2\delta\epsilon})\end{aligned}$$

for every 𝜖, η ∈ [0, 1] with 𝜖 + η = 1 which satisfies the energy estimate (5).

Proof

From Proposition 3 we know that

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle |\Lambda (x,\xi)|\leq M_{p-1}\left(\log2+\epsilon\log\langle x \rangle+\eta(p-1)\log\langle \xi \rangle_h\right)+\sum_{k=2}^{p-1}C_k M_{p-k} \\ & &\displaystyle \qquad \leq \delta \left(1+\epsilon\log\langle x \rangle+\eta(p-1)\log\langle \xi \rangle_h\right) \end{array} \end{aligned} $$

with a positive constant δ depending on M₁, …, M_p−1.This yelds

$$\displaystyle \begin{aligned} \begin{array}{rcl} |e^{\pm\Lambda(x,\xi)}|\leq e^{\delta}\langle x \rangle^{\delta\epsilon}\langle \xi \rangle_h^{\delta\eta(p-1)}, \end{array} \end{aligned} $$

and by the energy estimate (31) we get

for every t ∈ [0, T]. □

The next Proposition 5 states the regularity with respect to x, u of the coefficients a_j(t, x, ξ) of the linearized operator (7).

Proposition 5

Under the assumptions (5) and (6), there exists C′ > 0 such that for every fixed$u\in C([0,T]; H^{3p-1,3p-2}(\mathbb R))$the coefficients a_j(t, x, u(t, x)) of the operator P_u(D) satisfy for every 1 ≤ j ≤ p − 1, $(t,x)\in [0,T]\times \mathbb R$and β ≤ 3p − 2:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {}& &\displaystyle \qquad \ |\partial_x^\beta\mathop{\mathrm{Re}}\nolimits a_j(t,x,u(t,x))|\leq C'\gamma(u)(1+\|u\|{}_{1+\beta,\beta}^\beta)\langle x \rangle^{-\beta}, \end{array} \end{aligned} $$

(32)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {}& &\displaystyle \qquad \ |\partial_x^\beta\mathop{\mathrm{Im}}\nolimits a_j(t,x,u(t,x))|\leq C'\gamma(u)(1+\|u\|{}_{1+\beta,\beta}^\beta)\langle x\rangle^{-\frac{j}{p-1}-\beta}. \end{array} \end{aligned} $$

(33)

Proof

For every β ≥ 1 and 1 ≤ j ≤ p − 1 we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \partial_x^\beta( a_j(t,x,u))& =&\displaystyle (\partial_x^\beta a_j)(t,x,u) \\ & +&\displaystyle \displaystyle \sum_{\genfrac{}{}{0pt}{1}{\beta_1+\beta_2=\beta}{\beta_2\geq 1}}c_\beta \sum_{\genfrac{}{}{0pt}{1}{r_1+\ldots+r_{q}=\beta_2}{r_i\geq1}}c_{q,r} (\partial^q_w \partial_x^{\beta_1}a_j)(t,x,u)(\partial_x^{r_1}u)\cdots (\partial_x^{r_q}u) \end{array} \end{aligned} $$

for some c_β, c_q,r > 0. By (6), using the relationship between geometric and arithmetic mean value and Sobolev inequality, this gives for every β ≤ 4(p − 1):

where we have used the fact that for every 1 ≤ j ≤ q, β ≤ 3p − 2, we have

$$\displaystyle \begin{aligned}|\langle x \rangle^{r_j}\partial_x^{r_j}u|\leq C\|\langle x \rangle^{r_j}\partial_x^{r_j}u\|{}_{1,0}=\|u\|{}_{1+r_j,r_j}\leq \|u\|{}_{1+\beta,\beta}<\infty.\end{aligned}$$

On the other hand, looking at $\mathop {\mathrm {Im}} \nolimits a_j$ and using (5) instead of (6), the same computations give

$$\displaystyle \begin{aligned}|\partial_x^\beta(\mathop{\mathrm{Im}}\nolimits a_j(t,x,u))|\leq C''\gamma(u)(1+\|u\|{}^\beta_{1+\beta,\beta})\langle x \rangle^{-\frac j{p-1}-\beta}.\end{aligned}$$

□

Remark 5

We observe that a conjugation of the type $(e^{\lambda _{p-k}})^{-1}T_je^{\lambda _{p-k}}$ with λ_p−k given by (24) and T_j ∈SG^{j, 0}, j ≥ k + 1 depending on γ_j derivatives of u, by Proposition 4 gives:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hskip+0.5cm(e^{\lambda_{p-k}})^{-1}T_je^{\lambda_{p-k}}& =&\displaystyle e^{-\lambda_{p-k}}\left(T_j+r_{p-k}T_j\right) e^{\lambda_{p-k}} \end{array} \end{aligned} $$

(34)

where the principal symbol of r_p−k is given by ∂_ξλ_p−k(x, ξ)D_xλ_p−k(x, ξ) ∈SG^{−k, −(p−k)∕(p−1)}. By the asymptotic expansion we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma\left(T_j+r_{p-k}T_j\right)(x,\xi)=T_j(x,\xi)+\sum_{\alpha=0}^{j-k-1}\frac 1{\alpha!}\partial_\xi^\alpha r_{p-k}(x,\xi)D_x^\alpha T_j(x,\xi)+ S_0(x,\xi) \end{array} \end{aligned} $$

with S₀ ∈SG^{0, 0}. Since $\partial _\xi ^\alpha r_{p-k}D_x^\alpha T_j\in \mathbf {SG}^{j-k-\alpha , -(p-k)/(p-1)-|\alpha |}$ and depends on γ_j + α derivatives of u, by re-ordering the sum we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma\left(T_j+r_{p-k}T_j\right)(x,\xi)=T_j(x,\xi)+\sum_{\ell=1}^{j-k}T_{j,\ell}(x,\xi)+T_0 \end{array} \end{aligned} $$

with T_j,ℓ ∈SG^{ℓ, −(p−k)∕(p−1)−(j−k−ℓ)} depending on γ_j + j − k − ℓ derivatives of u and on M_p−k, T₀ of order (0, 0). Thus

$$\displaystyle \begin{aligned}(e^{\lambda_{p-k}})^{-1}\left(\displaystyle\sum_{j=0}^{p-1}T_j\right)e^{\lambda_{p-k}}=e^{-\lambda_{p-k}}\left(\sum_{j=0}^{p-1}( T_j+r_{p-k}T_j)\right) e^{\lambda_{p-k}}\end{aligned}$$

and we have, modulo terms of order (0, 0):

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle \sigma\left(\sum_{j=0}^{p-1}(T_j+r_{p-k}T_j)\right)(x,\xi)=\sum_{j=1}^{p-1}T_j(x,\xi)+\sum_{j=1}^{p-1}\sum_{\ell=1}^{j-k}T_{j,\ell}(x,\xi) \\ & &\displaystyle \qquad =\sum_{j=p-k}^{p-1}T_j+\sum_{j=1}^{p-k-1}\left(T_j+T_{j+k,j}+\ldots+T_{p-1,j}\right)=\sum_{j=1}^{p-1}T_j^{\prime} \end{array} \end{aligned} $$

with $T_j^{\prime }=T_j$ for j ≥ p − k, while for j ≤ p − k − 1 $T_j^{\prime } \in \mathbf {SG}^{j,0}$ as well as T_j but depend on $\max \{\gamma _{p-1}+p-1-k-j, \gamma _{p-2}+p-2-k-j,\ldots , \gamma _{j+k}\}$ derivatives of u and on the constant M_p−k.

Remark 6

Similarly, a conjugation of the type e^−λT_ke^λ, where λ ∈SG^{0, 0} and T_k ∈SG^{k, 0} depends on γ_k derivatives of u, gives, modulo terms of order (0, 0), the operator

$$\displaystyle \begin{aligned} \begin{array}{rcl} T_k+\sum_{\alpha=1}^{k-1}\frac 1{\alpha!}\left(\partial_\xi^\alpha T_k\right)e^{-\lambda}D_x^\alpha e^\lambda+\sum_{\beta=1}^{k-1}{\sum_{\alpha=0}^{k-\beta}\frac 1{\alpha!\beta!}\partial_\xi^\beta e^{-\lambda}D_x^\beta\left(\partial_\xi^\alpha T_kD_x^\alpha e^\lambda\right)}; \end{array} \end{aligned} $$

at each level 1 ≤ j ≤ k − 1 we find, except for T_j itself, new terms of type $\partial _\xi ^\beta e^{-\lambda }D_x^\beta \left (\partial _\xi ^\alpha T_{j+\alpha +\beta }D_x^\alpha e^\lambda \right )$ with the same decay as T_j and depending on γ_j+α+β + β derivatives of u.

Proof of Theorem 5

First of all we observe that the assumption (4) implies that a_p(t) ≥ C_p for every t ∈ [0, T] or a_p(t) ≤−C_p for every t ∈ [0, T] for a positive constant C_p. We will prove the theorem under the first condition. If the second one holds the result remains valid with only modifications of signs in the proof.

Fixed u, we consider the linear operator

$$\displaystyle \begin{aligned}iP_u(t,x,u(t,x), D_t,D_x)=\partial_t +ia_p(t)D_x^p+\displaystyle\sum_{j=0}^{p-1}ia_j(t,x,u) D_x^j\end{aligned}$$

with a_p satisfying (4) and a_j satisfying (32), (33) for every 1 ≤ j ≤ p − 1, and we apply for h ≥ h₁ (see Proposition 4) the first conjugation $(e^{\lambda _{p-1}})^{-1}iP_ue^{\lambda _{p-1}}$, with λ_p−1 in Definition 2 satisfying Proposition 3. Let us first notice that

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} & &\displaystyle (e^{\lambda_{p-1}})^{-1}iP_ue^{\lambda_{p-1}}= \partial_t +e^{-\lambda_{p-1}}\left(ia_p(t)D_x^p+\displaystyle\sum_{j=0}^{p-1}ia_j(t,x,u)D_x^j\right)e^{\lambda_{p-1}} \\ & &\displaystyle \qquad +e^{-\lambda_{p-1}}\left(ir_{p-1}(x,D)a_p(t)D_x^p+\displaystyle\sum_{j=0}^{p-1}ir_{p-1}(x,D)a_j(t,x,u)D_x^j\right) e^{\lambda_{p-1}} \end{array} \end{aligned} $$

and that the principal symbol of r_p−1 is given by ∂_ξλ_p−1(x, ξ)D_xλ_p−1(x, ξ) ∈SG^{−1, −1}. The composition $e^{-\lambda _{p-1}}ia_p\xi ^pe^{\lambda _{p-1}}$ provides, among others, the term − ∂_ξλ_p−1(x, ξ)a_pξ^p∂_xλ_p−1(x, ξ) = −ia_pξ^pr_p−1,−1(x, ξ) which cancels with the principal part of the symbol of $e^{-\lambda _{p-1}}ir_{p-1}a_p\xi ^pe^{\lambda _{p-1}}$. Then, we notice that by Remark 5 we can write

$$\displaystyle \begin{aligned} \begin{array}{rcl} (e^{\lambda_{p-1}})^{-1}iP_ue^{\lambda_{p-1}}& =&\displaystyle \partial_t +e^{-\lambda_{p-1}}\left(ia_p(t)D_x^p+\displaystyle\sum_{j=0}^{p-1}ia^{\prime}_j(t,x,u,D_x)\right)e^{\lambda_{p-1}} \\ & +&\displaystyle \mathrm{op} \left(ia_p\xi^pr_{p-1,-1}\right)(t,x,D) \end{array} \end{aligned} $$

with new terms

$$\displaystyle \begin{aligned}a^{\prime}_{p-1}(t,x,u,D_x)=a_{p-1}(t,x,u)D_x^{p-1}\end{aligned}$$

and, for 0 ≤ j ≤ p − 2, $a^{\prime }_j(t,x,u,D_x)$ is a pseudodifferential operator given by $a_j(t,x,u)D_x^j$ plus other terms of the same order. Namely, $a^{\prime }_j$ satisfy estimates of the form

$$\displaystyle \begin{aligned} \begin{array}{rcl} {}\ |\partial_\xi^\alpha\partial_x^\beta& &\displaystyle \mathop{\mathrm{Re}}\nolimits a_j^{\prime}(t,x,u(t,x),\xi)| \\ & \leq&\displaystyle C_{M_{p-1}}\gamma(u)(1+\|u\|{}_{p-1-j+\beta,p-2-j+\beta}^{p-2-j+\beta})\langle x \rangle^{-\beta}\langle \xi \rangle^{j-\alpha}, \end{array} \end{aligned} $$

(35)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {}\ |\partial_\xi^\alpha\partial_x^\beta& &\displaystyle \mathop{\mathrm{Im}}\nolimits a_j^{\prime}(t,x,u(t,x),\xi)| \\ & \leq&\displaystyle C_{M_{p-1}}\gamma(u)(1+\|u\|{}_{p-1-j+\beta,p-2-j+\beta}^{p-2-j+\beta})\langle x\rangle^{-\frac{j}{p-1}-\beta}\langle \xi \rangle^{j-\alpha}. \end{array} \end{aligned} $$

(36)

The asymptotic expansion gives

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} iP_1(t,x,u,D)& :=&\displaystyle (e^{\lambda_{p-1}})^{-1}iP_ue^{\lambda_{p-1}} \\ & =&\displaystyle \partial_t+ia_p(t)D_x^{p}+ia_{p-1}(t,x,u)D_x^{p-1} \\ & +&\displaystyle \mathrm{op}\left(ipa_p\xi^{p-1}D_x\lambda_{p-1}\right) \\ & +&\displaystyle \displaystyle \sum_{\alpha=2}^{p-1} \sum_{\stackrel{\beta+\gamma=\alpha}{\beta \leq p}} \frac 1{\beta! \gamma!} \mathrm{op}\left( a_p(t) \partial_\xi^\gamma e^{-\lambda_{p-1}} \cdot \partial_\xi^\beta \xi^p \cdot D_x^\alpha e^{\lambda_{p-1}}\right) \\ & +&\displaystyle \displaystyle\sum_{j=1}^{p-2}ia^{\prime}_j(t,x,u, D_x) \\ & +&\displaystyle \sum_{j=1}^{p-1}\sum_{\alpha=1}^{j-1}\frac 1{\alpha!}\mathrm{op}\left(e^{-\lambda_{p-1}}\partial_\xi^\alpha ia^{\prime}_jD_x^\alpha e^{\lambda_{p-1}}\right) \\ & +&\displaystyle \!\displaystyle\sum_{j=1}^{p-1}\sum_{\beta=1}^{j-1}\sum_{\alpha= 0}^{j-1-\beta}\sum_{\beta_1+\beta_2=\beta}\frac 1{\alpha!\beta_1!\beta_2!}\mathrm{op}\left(\!\partial_\xi^\beta e^{-\lambda_{p-1}}D_x^{\beta_1}\partial_\xi^\alpha ia^{\prime}_jD_x^{\alpha+\beta_2} e^{\lambda_{p-1}}\!\right) \\ & +&\displaystyle s_0(t,x,u,D) \end{array} \end{aligned} $$

(37)

with a term s₀ of order (0, 0). Notice that, by (35), (36) and Remark 6, in (37) we find at each level 1 ≤ j ≤ p − 2, except for the original terms $a_j(t,x,u)D_x^j$, terms with decay at least of type 〈x〉⁻¹, depending at most on M_p−1, and depending at most on

$$\displaystyle \begin{aligned}\gamma_{j+|\alpha|+|\beta|}+|\beta|= p-(j+|\alpha|+|\beta|)-1+|\beta|=p-j-|\alpha|-1\leq p-j-1\end{aligned}$$

derivatives of u, so that we get

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hskip+1cm iP_1& =&\displaystyle \partial_t+ia_p(t)D_x^p+ia_{p-1}(t,x,u)D_x^{p-1} \\ & &\displaystyle +\mathrm{op}\left(ipa_p\xi^{p-1}D_x\lambda_{p-1}\right) + \displaystyle\sum_{j=1}^{p-2}ia^{\prime\prime}_j(t,x,u, D_x)+ s_0(t,x,u,D) \end{array} \end{aligned} $$

(38)

where the pseudodifferential operators $a_j^{\prime \prime }$ are given by $a_jD_x^j$ plus other terms with the same behavior, namely $a_j^{\prime \prime }$ still satisfy (35) and (36).

Now, let us focus on the term A_p−1 of order p − 1 with respect to ξ in (38). By (24) and (33), the choice of ω in (25), and (4) we get for every |ξ|≥ 2h:

if we choose $M_{p-1}\geq \displaystyle\frac {C'\gamma (u)\sqrt {5}^{p-1}}{2^{p-1}pC_p}$, where we have also used the fact that $\displaystyle\frac {\langle x \rangle }{\langle \xi \rangle _h^{p-1}}\geq \frac 1 2$ on the support of $1-\psi \left (\displaystyle\frac {\langle x \rangle }{\langle \xi \rangle _h^{p-1}}\right )$ and $|\xi |{ }^{p-1} \geq (2/\sqrt {5})^{p-1}\langle \xi \rangle _h^{p-1}$ for |ξ|≥ 2h. Being the symbol $\mathop {\mathrm {Re}} \nolimits A_{p-1}(t,x,u,\xi )+2C'\gamma (u)$ non negative, we can apply the sharp Gårding Theorem 2 and we obtain that there exist pseudodifferential operators Q_p−1(t, x, u, D), R_p−1(t, x, u, D), R_0,p−1(t, x, u, D) with symbols

$$\displaystyle \begin{aligned} Q_{p-1}(t,x,u,\xi)\in \mathbf{SG}^{p-1,0},\quad R_{p-1}(t,x,u,\xi)\in \mathbf{SG}^{p-2,0},\quad R_{0,p-1}(t,x,u,\xi)\in S^0\end{aligned}$$

such that

$$\displaystyle \begin{aligned}A_{p-1}(t,x,u,D)=Q_{p-1}(t,x,u,D)+iR_{p-1}(t,x,u,D)+R_{0,p-1}(t,x,u, D)\end{aligned}$$

with

$$\displaystyle \begin{aligned}\mathop{\mathrm{Re}}\nolimits \langle Q_{p-1}(t,x,u,D)h(t,x), h(t,x)\rangle \geq 0 \quad \forall h\in \mathscr{S}(\mathbb R),\ (t,x)\in[0,T]\times\mathbb R\end{aligned}$$

and by (15)

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} R_{p-1}(t,x,u,\xi)& =&\displaystyle \sum_{j=1}^{p-2}R_{j,p-1} (t,x,u,\xi) \\ R_{p-2,p-1}& =&\displaystyle -i\left(\psi_1(\xi)D_xA_{p-1}+\!\!\!\!\!\!\displaystyle\sum_{2\leq \alpha+\beta\leq3}\psi_{\alpha,\beta}(\xi)\partial_\xi^\alpha D_x^\beta A_{p-1}\right) \\ R_{j,p-1}& =&\displaystyle -i \displaystyle\sum_{2(p-1-j)\leq \alpha+\beta\leq2(p-1-j)+1}\!\!\!\!\!\!\psi_{\alpha,\beta}(\xi)\partial_\xi^\alpha D_x^\beta A_{p-1} \end{array} \end{aligned} $$

(39)

for every 1 ≤ j ≤ p − 3, where ψ₁ and ψ_α,β are real valued symbols, ψ₁(ξ) ∈SG^{−1, 0} and ψ_α,β(ξ) ∈SG^{(α−β)∕2, 0}. We have so

$$\displaystyle \begin{aligned} \begin{array}{rcl} iP_1& =&\displaystyle \partial_t+ia_p(t)D_x^p+Q_{p-1}(t,x,u, D_x) \\ & +&\displaystyle iR_{p-1}(t,x,u, D_x)+\displaystyle\sum_{j=1}^{p-2}ia^{\prime\prime}_j(t,x,u, D_x)+ s_0(t,x,u, D_x). \end{array} \end{aligned} $$

We notice that, by (39), R_p−1 adds to the terms $a_j^{\prime \prime }$ some new terms; whenever β ≠ 0, these new terms have at least decay 〈x〉⁻¹, while for β = 0 we see that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathop{\mathrm{Re}}\nolimits\left(-i\psi_{\alpha,0}(\xi)\right.& &\displaystyle \left.\partial_\xi^\alpha A_{p-1}(t,x,u,\xi)\right) \\ & =&\displaystyle \psi_{\alpha,0}(\xi)\partial_\xi^\alpha \mathop{\mathrm{Im}}\nolimits A_{p-1}(t,x,u,\xi)\in \mathbf{SG}^{p-1-\alpha/2,0}\subset \mathbf{SG}^{p-2,0} \end{array} \end{aligned} $$

can be added to $\mathop {\mathrm {Re}} \nolimits a_j^{\prime \prime }$, while

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathop{\mathrm{Im}}\nolimits\left(-i\psi_{\alpha,0}(\xi)\right.& &\displaystyle \left.\partial_\xi^\alpha A_{p-1}(t,x,u,\xi)\right) \\ & =&\displaystyle -\psi_{\alpha,0}(\xi)\partial_\xi^\alpha\mathop{\mathrm{Re}}\nolimits A_{p-1}(t,x,u,\xi)\in \mathbf{SG}^{p-1-\alpha/2,-1}\subset \mathbf{SG}^{p-2,-\frac{p-2}{p-1}} \end{array} \end{aligned} $$

can be added to $\mathop {\mathrm {Im}} \nolimits a_j^{\prime \prime }$. Again, by (39), we see that the largest number of x-derivatives of u appears when α = 0, β = 2(p − 1 − j) + 1 and we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} |\psi_{\alpha,\beta}(\xi)\partial_\xi^\alpha D_x^\beta A_{p-1}(t,x,u,\xi)|& \leq &\displaystyle C'\gamma(u)\left(1+\|u\|{}_{\beta+1,\beta}^\beta\right)\langle \xi \rangle^{p-1-\frac{\alpha+\beta}2}\langle x \rangle^{-\beta} \\ & \leq &\displaystyle C'\gamma(u)\left(1+\|u\|{}_{2(p-j),2(p-j)-1}^{2(p-j)-1}\right)\langle \xi \rangle^{j}\langle x \rangle^{-1} \end{array} \end{aligned} $$

By these considerations, we understand that after the application of Theorem 2, we can write

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} iP_1=\partial_t+ia_p(t)D_x^p+Q_{p-1}(t,x,u, D_x)+\displaystyle\sum_{j=1}^{p-2}ia_{j,1}(t,x,u, D_x)+s_1(t,x,u,D) \end{array} \end{aligned} $$

(40)

for a new operator s₁ with symbol in S⁰, where a_j,1 are given by $a_j^{\prime \prime }$ plus other terms with the same order and decay, depending on 2(p − j) derivatives of u, this means that a_j,1 depend on $\max \{p-j-1,2(p-j)\}=2(p-j)$ derivatives of u. Summing up, for every β ≤ p − 1 (we need that 2(p − j) + β ≤ 2(p − 1) + β ≤ 3p − 1) we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} {}\ |\partial_\xi^\alpha\partial_x^\beta& &\displaystyle \mathop{\mathrm{Re}}\nolimits a_{j,1}(t,x,u(t,x),\xi)| \\ & \leq&\displaystyle C_{M_{p-1}}\gamma(u)(1+\|u\|{}_{2(p-j)+\beta,2(p-j)-1+\beta}^{2(p-j)-1+\beta})\langle x \rangle^{-\beta}\langle \xi \rangle^{j-\alpha}, \end{array} \end{aligned} $$

(41)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {}\ |\partial_\xi^\alpha\partial_x^\beta& &\displaystyle \mathop{\mathrm{Im}}\nolimits a_{j,1}(t,x,u(t,x),\xi)| \\ & \leq&\displaystyle C_{M_{p-1}}\gamma(u)(1+\|u\|{}_{2(p-j)+\beta,2(p-j)-1+\beta}^{2(p-j)-1+\beta})\langle x\rangle^{-\frac{j}{p-1}-\beta}\langle \xi \rangle^{j-\alpha}. \end{array} \end{aligned} $$

(42)

Now, let us consider, for $h\geq \max \{h_1,h_2\}$ (see Proposition 4), the operator $(e^{\lambda _{p-2}})^{-1}iP_1e^{\lambda _{p-2}}$, with λ_p−2 in Definition 2 satisfying Proposition 3. We observe preliminarly that, since $e^{\pm \lambda _{p-2}} \in \mathbf {SG}^{0,0}(\mathbb R^2) \subset S^0(\mathbb R^2)$, then for the composition $(e^{\lambda _{p-2}})^{-1}s_1(t,x,u,D) e^{\lambda _{p-2}}$ we can use the symbolic calculus in the Hörmander class and obtain that $(e^{\lambda _{p-2}})^{-1}s_1(t,x,u,D) e^{\lambda _{p-2}}$ is again an operator with symbol in $S^0(\mathbb R^2).$ Moreover, since $\left (e^{\lambda _{p-2}}\right )^{-1}=e^{-\lambda _{p-2}}(I+r_{p-2})$ and the principal part of r_p−2 has symbol r_p−2,−2(x, ξ) = ∂_ξλ_p−2(x, ξ)D_xλ_p−2(x, ξ) in $\mathbf {SG}^{-2,-\frac {p-2}{p-1}}$, by Remark 5 we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle (e^{\lambda_{p-2}})^{-1}iP_1e^{\lambda_{p-2}}= \partial_t+\mathrm{op} \left(ia_pr_{p-2,-2}\right) \\ & &\displaystyle \qquad + e^{-\lambda_{p-2}}\left(ia_p(t)D_x^p+Q_{p-1}(t,x,u,D)+\displaystyle\sum_{j=0}^{p-2}ia^{\prime}_{j,1}(t,x,u, D_x) +s_1(t,x,u,D)\right)e^{\lambda_{p-2}} \end{array} \end{aligned} $$

with $a^{\prime }_{p-2,1}(t,x,u,D_x)=a_{p-2,1}(t,x,u,D_x)$ and, for 0 ≤ j ≤ p − 3, $a^{\prime }_{j,1}(t,x,u, D_x)$ is given by a_j,1(t, x, u, D_x) plus some new terms with the same order and decay as a_j,1 and depending on $\max \{\gamma _{p-1}+p-1-2-j,\ldots , \gamma _{p-\ell }+p-\ell -2-j,\ldots \gamma _{j+2}\}=\gamma _{j+2}=2(p-j-2)$, because we have γ_p−ℓ = 2(p − (p − ℓ)) = 2ℓ for 1 ≤ ℓ ≤ p − 1. The new terms contain a smaller number of derivatives with respect to (41) and (42). Thus for every 1 ≤ j ≤ p − 2 we have that $a_{j,1}^{\prime }$ still satisfy (41) and (42) for a constant depending also on M_p−2; notice that the dependence on M_p−2 is only at levels 1 ≤ j ≤ p − 3. The asymptotic expansion gives

$$\displaystyle \begin{aligned} \begin{array}{rcl} iP_2(t,x,u,D)& :=&\displaystyle (e^{\lambda_{p-2}})^{-1}iP_1e^{\lambda_{p-2}} \\{} & =&\displaystyle \partial_t+ia_p(t)D_x^{p}+Q_{p-1}(t,x,u, D) \\ & +&\displaystyle ia_{p-2,1}(t,x,u, D_x)+\mathrm{op}\left(ipa_p\xi^{p-1}D_x\lambda_{p-2}\right) \\ & +&\displaystyle \displaystyle \sum_{\beta =2}^{p-1}\frac 1{\beta!} \mathrm{op}\left( \partial_\xi^\beta(ia_p \xi^p e^{-\lambda_{p-2}}) D_x^\beta \lambda_{p-2} \right) \\ & +&\displaystyle \displaystyle\sum_{j=1}^{p-3}ia^{\prime}_{j,1}(t,x,u, D_x) + \displaystyle\sum_{\alpha=1}^{p-2}\frac 1{\alpha!}\mathrm{op}\left(e^{-\lambda_{p-2}}\partial_\xi^\alpha Q_{p-1}D_x^\alpha e^{\lambda_{p-2}}\right) \\ & +&\displaystyle \sum_{\beta= 1}^{p-2}\displaystyle\sum_{\alpha=0}^{p-2-\beta}\sum_{\beta_1+\beta_2=\beta}\frac 1{\alpha!\beta_1!\beta_2!}\mathrm{op}\left(\partial_\xi^\beta e^{-\lambda_{p-2}}D_x^{\beta_1}\partial_\xi^\alpha Q_{p-1}D_x^{\alpha+\beta_2} e^{\lambda_{p-2}}\right) \\ & +&\displaystyle \displaystyle\sum_{j=1}^{p-2}\sum_{\alpha=1}^{j-1}\frac 1{\alpha!}\mathrm{op}\left(e^{-\lambda_{p-2}}\partial_\xi^\alpha ia^{\prime}_{j,1}D_x^\alpha e^{\lambda_{p-2}}\right) \\ & +&\displaystyle \displaystyle\sum_{j=1}^{p-2}\sum_{\beta= 1}^{j-1}\sum_{\alpha= 0}^{j-1-\beta}\sum_{\beta_1+\beta_2=\beta}\frac 1{\alpha!\beta_1!\beta_2!}\mathrm{op}\left(\partial_\xi^\beta e^{-\lambda_{p-2}}D_x^{\beta_1}\partial_\xi^\alpha ia^{\prime}_{j,1}D_x^{\alpha+\beta_2} e^{\lambda_{p-2}}\right) \\ & +&\displaystyle s^{\prime}_1(t,x,u,D) \end{array} \end{aligned} $$

(43)

with a new term $s^{\prime }_1 \in S^0$. Let us now look at (43); by (41), (42), and using the estimate (28) wih k = 2, we find at each level 1 ≤ k ≤ p − 3, the original terms a_k,1(t, x, u, D) plus terms which decay with respect to x at least like 〈x〉⁻¹, and possibly depending only on M_p−1 and M_p−2; the largest number of derivatives with respect to u appears in

$$\displaystyle \begin{aligned} \begin{array}{rcl} |\partial_\xi^\beta e^{-\lambda_{p-2}}& &\displaystyle D_x^{\beta_1}\partial_\xi^\alpha ia^{\prime}_{j,1}D_x^{\alpha+\beta_2} e^{\lambda_{p-2}}| \\ & \leq&\displaystyle C^{\prime}_{M_{p-1},M_{p-2}}\gamma(u)(1+\|u\|{}_{2(p-j)+\beta,2(p-j)+\beta-1}^{2(p-j)+\beta-1})\langle \xi \rangle^{j-\alpha-\beta}\langle x \rangle^{-\alpha-\beta}; \end{array} \end{aligned} $$

at the level k = j − α − β the largest number of x-derivatives of u appears when α = 0 and β = j − k and it is given by 2(p − j) + β = 2(p − k − β) + β = 2(p − k) − β ≤ 2(p − k) − 1. Thus, similarly as for (38), we get

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hskip+1cm iP_2& =&\displaystyle \partial_t+ia_p(t)D_x^p+Q_{p-1}(t,x,u, D)+ia_{p-2,1}(t,x,u, D_x) \\ & +&\displaystyle \mathrm{op}\left(ip a_p\xi^{p-1}D_x\lambda_{p-2}\right)+\displaystyle\sum_{j=1}^{p-3}ia^{\prime\prime}_{j,1}(t,x,u, D_x)+ s^{\prime}_1(t,x,u,D) \end{array} \end{aligned} $$

(44)

where $a_{j,1}^{\prime \prime }$ are given by a_j plus other terms of the same type, still satisfying (41), (42) but with a constant $C_{M_{p-1},M_{p-2}}$ depending on both M_p−1 and M_p−2.

Now, let us focus on the term A_p−2 of order p − 2 with respect to ξ in (44). By (42), (24), the choice of ω in (25), and (4) we get for every |ξ|≥ 2h:

if we choose $M_{p-2}\geq \displaystyle\frac {C_{M_{p-1}}\gamma (u)\left (1+\|u\|{ }_{4,3}^3\right )\sqrt {5}^{p-1}}{2^{p-1}pC_p}$, and using again $\langle x \rangle /\langle \xi \rangle _h^{p-1}\geq 1/2$ on the support of $1-\psi (\langle x \rangle /\langle \xi \rangle _h^{p-1})$ and $|\xi |{ }^p \geq (2/\sqrt {5})^{p-1}\langle \xi \rangle _h^{p-1}$ for |ξ|≥ 2h. We can so apply the sharp Gårding theorem to the symbol $A_{p-2}(t,x,u,\xi )+2C_{M_{p-1}}\gamma (u)\left (1+\|u\|{ }_{4,3}^3\right )\geq 0$ and we obtain that there exist pseudodifferential operators Q_p−2(t, x, u, D), R_p−2(t, x, u, D), R_0,p−2(t, x, u, D) with symbols

$$\displaystyle \begin{aligned} Q_{p-2}(t,x,u,\xi)\in \mathbf{SG}^{p-2,0},\quad R_{p-2}(t,x,u,\xi)\in \mathbf{SG}^{p-3,0},\quad R_{0,p-2}(t,x,u,\xi)\in S^0\end{aligned}$$

such that

$$\displaystyle \begin{aligned}A_{p-2}(t,x,u,D)=Q_{p-2}(t,x,u,D)+iR_{p-2}(t,x,u,D)+R_{0,p-2}(t,x,uD)\end{aligned}$$

with

$$\displaystyle \begin{aligned}\mathop{\mathrm{Re}}\nolimits \langle Q_{p-2}(t,x,u,D)h(t,x), h(t,x)\rangle \geq 0 \quad \forall h\in \mathscr{S}(\mathbb R),\ (t,x)\in[0,T]\times\mathbb R\end{aligned} $$

and

$$\displaystyle \begin{aligned} R_{p-2}=\sum_{j=1}^{p-3}R_{j,p-2}\end{aligned} $$

(45)

where

$$\displaystyle \begin{aligned}R_{p-3,p-2}= -i\left(\psi_1(\xi)D_xA_{p-2}+\!\!\!\displaystyle\sum_{2\leq \alpha+\beta\leq3}\psi_{\alpha,\beta}(\xi)\partial_\xi^\alpha D_x^\beta A_{p-2}\right)\end{aligned} $$

and

$$\displaystyle \begin{aligned}R_{j,p-2}=-i\displaystyle\sum_{2(p-2-j)\leq \alpha+\beta\leq2(p-2-j)+1}\!\!\!\psi_{\alpha,\beta}(\xi)\partial_\xi^\alpha D_x^\beta A_{p-2},\end{aligned} $$

for every 1 ≤ j ≤ p − 3. We have so

$$\displaystyle \begin{aligned} \begin{array}{rcl} iP_2& =&\displaystyle \partial_t+ia_p(t,D)+Q_{p-1}(t,x,u, D_x)+Q_{p-2}(t,x,u, D_x) \\ & +&\displaystyle iR_{p-2}(t,x,u, D_x)+\displaystyle\sum_{j=1}^{p-3}ia^{\prime\prime}_{j,1}(t,x,u, D_x)+ s^{\prime\prime}_{1}(t,x,u, D_x). \end{array} \end{aligned} $$

Again, each R_j,p−2 adds to $a_{j,1}^{\prime \prime }$ new terms with the same order and decay as $a_{j,1}^{\prime \prime }$ (notice that the second application of Theorem 2 is needed only in the case p ≥ 3 and in this case we have 5 ≤ p + 2, so the term ψ₁(ξ)D_xA_p−2(t, x, u, ξ) satisfies (41) and (42) with j = p − 3 and a constant depending on M_p−1, M_p−2. The largest number of x-derivatives of u appears when α = 0, β = 2(p − 2 − j) + 1 and we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} |\psi_{\alpha,\beta}(\xi)\partial_\xi^\alpha D_x^\beta A_{p-2}(t,x,u,\xi)|& \leq &\displaystyle C_{M_{p-1}}\gamma(u)\left(\!1+\|u\|{}_{4+\beta,3+\beta}^{3+\beta}\!\right)\langle \xi \rangle^{p-2-\frac{\alpha+\beta}2}\langle x \rangle^{-\beta} \\ & \leq &\displaystyle C_{M_{p-1}}\gamma(u)\left(\!1+\|u\|{}_{2(p-j)+1,2(p-j)}^{2(p-j)}\!\right)\langle \xi \rangle^{j}\langle x \rangle^{-1}. \end{array} \end{aligned} $$

This means that, after the second application of the sharp Gårding theorem, we can write

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hskip+1cm iP_2& =&\displaystyle \partial_t+ia_p(t,D)+Q_{p-1}(t,x,u, D_x)+Q_{p-2}(t,x,u, D_x) \\ & +&\displaystyle \displaystyle\sum_{j=1}^{p-3}ia_{j,2}(t,x,u, D_x)+s_2(t,x,u,D) \end{array} \end{aligned} $$

(46)

for a new operator s₂ with symbol in S⁰, where a_j,2 are given by $a_jD_x^j$ plus other terms with the same order and decay depending on 2(p − j) + 1 x-derivatives of u; thus a_j,2 depends on $\max \{2(p-j)+1,2(p-j)\}=2(p-j)+1$x-derivatives of u. Summing up, for every 1 ≤ j ≤ p − 3 and for β ≤ p (we need that 2(p − j) + 1 + β ≤ 2p − 1 + β ≤ 3p − 1) we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \ |\partial_\xi^\alpha\partial_x^\beta&\displaystyle &\displaystyle {\mathrm{Re}} a_{j,2}(t,x,u(t,x),\xi)| \\ &\displaystyle \leq&\displaystyle C_{M_{p-1},M_{p-2}}\gamma(u)(1+\|u\|{}_{2(p-j)+1+\beta,2(p-j)+\beta}^{2(p-j)+\beta})\langle x \rangle^{-\beta}\langle \xi \rangle^{j-\alpha}, {} \end{array} \end{aligned} $$

(47)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \ |\partial_\xi^\alpha\partial_x^\beta&\displaystyle &\displaystyle {\mathrm{Im}} a_{j,2}(t,x,u(t,x),\xi)| \\ &\displaystyle \leq&\displaystyle C_{M_{p-1},M_{p-2}}\gamma(u)(1+\|u\|{}_{2(p-j)+1+\beta,2(p-j)+\beta}^{2(p-j)+\beta})\langle x\rangle^{-\frac{j}{p-1}-\beta}\langle \xi \rangle^{j-\alpha}.{} \end{array} \end{aligned} $$

(48)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & \leq&\displaystyle C_{M_{p-1},M_{p-2}}\gamma(u)(1+\|u\|{}_{2(p-j)+1+\beta,2(p-j)+\beta}^{2(p-j)+\beta})\langle x\rangle^{-\frac{j}{p-1}-\beta}\langle \xi \rangle^{j-\alpha}. \end{array} \end{aligned} $$

We can proceed performing the next conjugations which follow the same argument as the second one. Arguing in this way, after ℓ = p − 3 applications of Theorem 2 we finally come for $h\geq \max \{h_1,\ldots ,h_{p-3}\}$ to

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \hskip+1cm iP_{p-3}& =&\displaystyle (e^{\lambda_3})^{-1}\ldots (e^{\lambda_{p-1}})^{-1} (iP)(e^{\lambda_{p-1}})\ldots (e^{\lambda_3}) \end{array} \end{aligned} $$

(49)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle = \partial_t+ia_p(t)D_x^p+Q_{p-1}(t,x,u, D_x)+\ldots+Q_{3}(t,x,u, D_x) \quad \\ & +&\displaystyle ia_{2,p-3}(t,x,u, D_x)+ia_{1,p-3}(t,x,u, D_x)+s_{p-3}(t,x,u,D) \end{array} \end{aligned} $$

(50)

where, for every 1 ≤ j ≤ p − 3,

$$\displaystyle \begin{aligned}\mathop{\mathrm{Re}}\nolimits \langle Q_{p-j}(t,x,u,D)h(t,x), h(t,x)\rangle \geq 0 \quad \forall h\in \mathscr{S}(\mathbb R),\ (t,x)\in[0,T]\times\mathbb R\end{aligned}$$

and moreover for every β ≤ 7

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} |\partial_\xi^\alpha\partial_x^\beta& &\displaystyle \mathop{\mathrm{Re}}\nolimits a_{2,p-3}(t,x,u,\xi)| \end{array} \end{aligned} $$

(51)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & \leq&\displaystyle C_{M_{p-1},\ldots, M_{3}}\gamma(u)(1+\|u\|{}_{3p-8+\beta,3p-9+\beta}^{3p-9+\beta})\langle x \rangle^{-\beta}\langle \xi \rangle^{2-\alpha}, \\ {} |\partial_\xi^\alpha\partial_x^\beta& &\displaystyle \mathop{\mathrm{Im}}\nolimits a_{2,p-3}(t,x,u,\xi)| \\ & \leq&\displaystyle C_{M_{p-1},\ldots, M_{3}}\gamma(u)(1+\|u\|{}_{3p-8+\beta,3p-9+\beta}^{3p-9+\beta})\langle x \rangle^{-\frac{2}{p-1}-\beta}\langle \xi \rangle^{2-\alpha}, \end{array} \end{aligned} $$

(52)

and for β ≤ 5

$$\displaystyle \begin{aligned} \begin{array}{rcl} {}|\partial_\xi^\alpha\partial_x^\beta& &\displaystyle \mathop{\mathrm{Re}}\nolimits a_{1,p-3}(t,x,u,\xi)| \end{array} \end{aligned} $$

(53)

$$\displaystyle \begin{aligned} \begin{array}{rcl} & \leq&\displaystyle C_{M_{p-1},\ldots, M_{3}}\gamma(u)(1+\|u\|{}_{3p-6+\beta,3p-7+\beta}^{3p-7+\beta})\langle x \rangle^{-\beta}\langle \xi \rangle^{1-\alpha}, \\ {}|\partial_\xi^\alpha\partial_x^\beta& &\displaystyle \mathop{\mathrm{Im}}\nolimits a_{1,p-3}(t,x,u,\xi)| \\ & \leq&\displaystyle C_{M_{p-1},\ldots, M_{3}}\gamma(u)(1+\|u\|{}_{3p-6+\beta,3p-7+\beta}^{3p-7+\beta})\langle x \rangle^{-\frac{1}{p-1}-\beta}\langle \xi \rangle^{1-\alpha}. \end{array} \end{aligned} $$

(54)

Now, we define, for $h\geq \max \{h_1,\ldots ,h_{p-2}\}$, $iP_{p-2}(t,x,u,D):=(e^{\lambda _{2}})^{-1}iP_{p-3}e^{\lambda _{2}}$ and we get

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} iP_{p-2}& =&\displaystyle \partial_t+ia_p(t)D_x^p+Q_{p-1}(t,x,u, D_x)+\ldots+Q_{3}(t,x,u, D_x) \\ & +&\displaystyle ia_{2,p-3}(t,x,u, D_x)+\mathrm{op}\left(ipa_p\xi^{p-1}D_x\lambda_{2}\right) \\ & +&\displaystyle ia_{1,p-3}^{\prime\prime}(t,x,u, D_x)+s_{p-3}^{\prime}(t,x,u,D) \end{array} \end{aligned} $$

(55)

where $a_{1,p-3}^{\prime \prime }$ are given by a_j plus other terms of the same type, still satisfying (53) and (54) but with a constant $C_{M_{p-1},\ldots ,M_{2}}$ instead of $C_{M_{p-1},\ldots ,M_{3}}$, and $s_{p-3}^{\prime }$ is still of order 0.

Now, as usual, by choosing $M_2\geq \displaystyle C_{M_{p-1}, \ldots , M_{3}}\gamma (u)\left (1+\|u\|{ }_{3p-8,3p-9}^{3p-9}\right )$$\frac {\sqrt {5}^{p-1}}{2^{p-1}pC_p}$ we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathop{\mathrm{Re}}\nolimits A_2:& =&\displaystyle \mathop{\mathrm{Re}}\nolimits\left(ia_{2,p-3}(t,x,u, D_x)+\mathrm{op}\left(pa_p\xi^{p-1}\partial_x\lambda_{2}\right) \right) \\ & \geq&\displaystyle 2C_{M_{p-1}, \ldots, M_{3}}\gamma(u)\left(1+\|u\|{}_{3p-8,3p-9}^{3p-9}\right). \end{array} \end{aligned} $$

This time, since we are dealing with second order operators, we can apply the Fefferman-Phong inequality (see Theorem 3) to

$$\displaystyle \begin{aligned}\mathop{\mathrm{Re}}\nolimits A_2+2C_{M_{p-1}, \ldots, M_{3}}\gamma(u)\left(1+\|u\|{}_{3p-8,3p-9}^{3p-9}\right)\end{aligned}$$

and obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathop{\mathrm{Re}}\nolimits\langle \mathop{\mathrm{Re}}\nolimits A_2h, h\rangle\geq -\left(c+2C_{M_{p-1}, \ldots, M_{3}}\gamma(u)\left(1+\|u\|{}_{3p-8,3p-9}^{3p-9}\right)\right)\|h\|{}^2,\quad \forall h\in \mathscr{S}(\mathbb R) \end{array} \end{aligned} $$

for a positive constant c = c(u) depending on the derivatives $\partial _\xi ^\alpha \partial _x^\beta$ with |α| + |β|≤ 7 of the symbol $\mathop {\mathrm {Re}} \nolimits A_2(t,x,u,\xi )+2C_{M_{p-1}, \ldots , M_{3}}\gamma (u)\left (1+\|u\|{ }_{3p-8,3p-9}^{3p-9}\right )$. Since γ is of class C⁷, we can now find a constant C_γ > 0 (depending also on M_p−1, …, M₃) such that

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathop{\mathrm{Re}}\nolimits\langle \mathop{\mathrm{Re}}\nolimits A_2h, h\rangle& \geq&\displaystyle -C_\gamma\left(1+\|u\|{}_{3p-8+7,3p-9+7}^{3p-9+7}\right)\|h\|{}^2 \\ & =&\displaystyle -C_\gamma\left(1+\|u\|{}_{3p-1,3p-2}^{3p-2}\right)\|h\|{}^2,\quad \forall h\in \mathscr{S}(\mathbb R) . \end{array} \end{aligned} $$

The advantage of the use of Fefferman-Phong inequality instead of Theorem 2 is that we avoid the remainder of that theorem, i.e. we save some derivatives of the fixed function u.

It now remains to treat the terms $i \mathop {\mathrm {Im}} \nolimits A_{2}=i \mathop {\mathrm {Re}} \nolimits a_{2,p-3}$ and $ia_{1,p-3}^{\prime \prime }$ in (55). We split $i \mathop {\mathrm {Re}} \nolimits a_{2,p-3}$ into its Hermitian and anti-Hermitian part:

$$\displaystyle \begin{aligned}i\mathop{\mathrm{Im}}\nolimits A_2=\frac{i\mathop{\mathrm{Re}}\nolimits a_{2,p-3}+(i\mathop{\mathrm{Re}}\nolimits a_{2,p-3})^*}{2}+\frac{i\mathop{\mathrm{Re}}\nolimits a_{2,p-3}- (i\mathop{\mathrm{Re}}\nolimits a_{2,p-3})^*}{2}=:H_1+H_2,\end{aligned}$$

and we have that

$$\displaystyle \begin{aligned}\mathop{\mathrm{Re}}\nolimits\langle H_2h, h\rangle=0,\end{aligned}$$

while

$$\displaystyle \begin{aligned}H_1=-\frac 12\partial_\xi\partial_x\mathop{\mathrm{Re}}\nolimits a_{2,p-3}\ (\rm{mod.}\ \mathbf{SG}^{0,0})\end{aligned}$$

can be put together with $ia_{1,p-3}^{\prime \prime }$ since by (51) it satisfies (53). We get so

$$\displaystyle \begin{aligned} \begin{array}{rcl} \hskip+1cm iP_{p-2}& =&\displaystyle \partial_t+ia_p(t)D_x^p+Q_{p-1}(t,x,u, D_x)+\ldots+Q_{3}(t,x,u, D_x) \\ & +&\displaystyle \mathop{\mathrm{Re}}\nolimits A_2(t,x,u,D_x)+ H_2(t,x,u,D_x) \\ & +&\displaystyle ia_{1,p-2}(t,x,u, D_x)+s_{p-2}(t,x,u,D) \end{array} \end{aligned} $$

with ia_1,p−2 still satisfying (53), (54) and s_p−2 ∈ S⁰. Finally, to treat the terms of order 1 with respect to ξ, we perform for $h\geq \max \{h_1,\ldots ,h_{p-1}\}$ the last conjugation:

$$\displaystyle \begin{aligned} \begin{array}{rcl}{} iP_{\Lambda}:& =&\displaystyle (e^{\lambda_{1}})^{-1}iP_{p-2}e^{\lambda_1} \\ & =&\displaystyle \partial_t+ia_p(t)D_x^p+Q_{p-1}(t,x,u, D_x)+\ldots+Q_{3}(t,x,u, D_x) \\ & +&\displaystyle \mathop{\mathrm{Re}}\nolimits A_2(t,x,u,D_x)+ H_2(t,x,u,D_x) \\ & +&\displaystyle ia_{1,p-2}(t,x,u, D_x)+\mathrm{op}\left(ipa_p\xi^{p-1}D_x\lambda_{1}\right) +s_{p-2}^{\prime}(t,x,u,D) \end{array} \end{aligned} $$

(56)

with a new term $s^{\prime }_{p-2} \in S^0$. Notice that the conjugation $e^{-\lambda _{1}}\left ( \mathop {\mathrm {Re}} \nolimits A_2+ H_2\right )e^{\lambda _1}$ gives $\mathop {\mathrm {Re}} \nolimits A_2+ H_2$ plus a remainder of order (0, 0) whose principal part is given by

$$\displaystyle \begin{aligned}\partial_\xi (\mathop{\mathrm{Re}}\nolimits A_{2}+H_2)\partial_x\lambda_1-\partial_\xi\lambda_1D_x(\mathop{\mathrm{Re}}\nolimits A_2+ H_2)-\partial_\xi\lambda_1(\mathop{\mathrm{Re}}\nolimits A_2+ H_2)D_x\lambda_1\in \mathbf{SG}^{0,0}.\end{aligned} $$

As usual, by choosing $M_1\geq \displaystyle C_{M_{p-1}, \ldots , M_{2}}\gamma (u)\left (1+\|u\|{ }_{3p-6,3p-7}^{3p-7}\right )\sqrt {5}^{p-1}/$$\left (2^{p-1}pC_p\right )$ we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathop{\mathrm{Re}}\nolimits A_1:& =&\displaystyle \mathop{\mathrm{Re}}\nolimits\left(ia_{1,p-2}(t,x,u, D_x)+\mathrm{op}\left(pa_p\xi^{p-1}\partial_x\lambda_{1}\right)\right) \\ & \geq&\displaystyle 0- 2C_{M_{p-1}, \ldots, M_{2}}\gamma(u)\left(1+\|u\|{}_{3p-6,3p-7}^{3p-7}\right). \end{array} \end{aligned} $$

To the symbol A₁(t, x, u, ξ) we can apply the sharp Gårding inequality (16) and we obtain

$$\displaystyle \begin{aligned}\mathop{\mathrm{Re}}\nolimits\langle A_1h, h\rangle\geq -C^{\prime}_\gamma (1+\|u\|{}_{3p-6,3p-7}^{3p-7})\|h\| \qquad \forall h\in \mathscr{S}(\mathbb R).\end{aligned}$$

At this point we are finally ready to prove an energy estimate in L² for the Cauchy problem. We compute

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac d{dt}\|w(t)\|{}^2& =&\displaystyle 2\mathop{\mathrm{Re}}\nolimits\langle \partial_tw,w\rangle=2\mathop{\mathrm{Re}}\nolimits\langle iP_\Lambda w,w\rangle- 2\mathop{\mathrm{Re}}\nolimits\langle ia_pw,w\rangle -\sum_{k=3}^{p-1}2\mathop{\mathrm{Re}}\nolimits\langle Q_kw,w\rangle \\ & -&\displaystyle 2\mathop{\mathrm{Re}}\nolimits\langle \mathop{\mathrm{Re}}\nolimits A_2w,w\rangle-2\mathop{\mathrm{Re}}\nolimits\langle H_2w,w\rangle- 2\mathop{\mathrm{Re}}\nolimits\langle A_1w,w\rangle- 2\mathop{\mathrm{Re}}\nolimits\langle s^{\prime}_{p-2}w,w\rangle \\ & \leq &\displaystyle C^{\prime}_\gamma (1+\|u\|{}_{3p-1,3p-2}^{3p-2})\left(\|P_\Lambda w\|{}^2+\|w\|{}^2\right)\quad \forall w\in \mathscr{S}(\mathbb R). \end{array} \end{aligned} $$

By Gronwall’s Lemma we obtain

$$\displaystyle \begin{aligned} \begin{array}{rcl} \|w\|{}^2\leq C e^{(1+\|u\|{}_{3p-1,3p-2}^{3p-2})t}\left(\|w(0,\cdot)\|{}^2+\int_0^t \|P_\Lambda w(\tau, \cdot)\|{}^2d\tau\right) \end{array} \end{aligned} $$

and, by standard arguments, the energy estimate (31). □

Remark 7

Notice that with respect to [1], by a different proof we can relax from 4p − 3 to 3p − 1 the number of derivatives of u needed to perform the computations in the linearized problem.

4 The Semilinear Problem

We now apply the energy estimates obtained in the previous section to prove the well posedness of the semilinear Cauchy problem (1). Fixed s₂ ≥ 3p − 2 and T > 0, we consider the space $X_T^{s_2}:= C^1([0,T], H^{\infty ,s_2}(\mathbb R))$ and the map $J: X_T^{s_2} \to X_T^{s_2}$ defined by

$$\displaystyle \begin{aligned} \begin{array}{rcl} J(u) & :=&\displaystyle u(t,x) - u_0(x) +i \int_0^t a_p(t)D_x^p u(s,x)\, ds \\ & &\displaystyle +i \sum_{j=0}^{p-1}\int_0^t a_j(s,x,u(s,x)) D_x^j u(s,x)\, ds -i \int_0^t f(s,x)\, ds. \end{array} \end{aligned} $$

It is well known that the existence of a unique solution of (1) in $X_{T^*}^{s_2}$ for some T^∗∈ (0, T] is equivalent to the existence of a unique solution in $X_{T^*}^{s_2}$ of the equation Ju = 0, cf. [1, 12]. We shall approach the latter problem via the Nash-Moser inversion theorem. As a direct consequence of Lemma 1, $X_T^{s_2}$ is a tame Fréchet space endowed with the family of seminorms

$$\displaystyle \begin{aligned}|g|{}_{n,s_2,T} = \sup_{[0,T]}(|g(t, \cdot)|{}_{n,s_2} + |D_tg(t, \cdot)|{}_{n,s_2}), \qquad n =0,1,2,\ldots\end{aligned}$$

The map J is smooth tame since it is defined in terms of sums and composition of integration and linear and nonlinear partial differential operators. In order to apply Nash-Moser theorem it is sufficient to prove that for every fixed $u,h \in X_T^{s_2}$, the equation DJ(u)v = h has a unique solution $v=S(u,h) \in X_T^{s_2}$ and that the map

$$\displaystyle \begin{aligned} \begin{array}{rcl}{}S: X_T^{s_2} \times X_T^{s_2} \to X_T^{s_2},\quad (u,h)\to v=S(u,h) \end{array} \end{aligned} $$

(57)

is smooth tame.

Lemma 3

For every$u,h \in X_T^{s_2},$the equation DJ(u)v = h admits a unique solution$v \in X_T^{s_2}$satisfying for every$n \in \mathbb N$the following estimate:

$$\displaystyle \begin{aligned}| v(t, \cdot)|{}_{n,s_2}^2 \leq C_n(u) \left( | h(0, \cdot)|{}^2_{n+r,s_2} + \int_0^t |D_t h(\tau, \cdot)|{}_{n+r,s_2}^2\, d\tau \right), \qquad t \in [0,T] \end{aligned} $$

(58)

for every r ≥ 2δ(p − 1) with$C_n(u)=C_{n+2\delta (p-1),\gamma }\exp ( 1+\| u\|{ }^{3p-2}_{3p-1, 3p-2})$.

Proof

The proof follows the same argument as the proof of [1, Lemma 3.2], so we just sketch it. A direct computation of the Fréchet derivative of J gives

$$\displaystyle \begin{aligned} \begin{array}{rcl} DJ(u)v & :=&\displaystyle \lim_{\varepsilon \to 0} \frac{J(u+\varepsilon v)-J(u)}{\varepsilon} \\ & =&\displaystyle v +i \int_0^t a_p(s)D_x^p v(s)\, ds + i \sum_{j=0}^{p-1} \int_0^t \tilde{a}_j(s,x,u) D_x^j v(s)\, ds, \end{array} \end{aligned} $$

where

$$\displaystyle \begin{aligned}\tilde{a}_j(s,x,u)= \begin{cases} a_j(s,x,u) \qquad 1 \leq j \leq p-1 \\ a_0(s,x,u) + \sum_{h=0}^{p-1}\partial_w a_h(s,x,u) D_x^h u, \qquad j=0. \end{cases}\end{aligned}$$

Hence v is a solution of the equation DJ(u)v = h if and only if it is a solution of the equation $J_{h_0,u,D_t h}(v)=0$, where h₀(x) := h(0, x) and for every $u,u_0,f \in X_T^{s_2}$ the map $J_{u_0,u,f}: X_T^{s_2} \to X_T^{s_2}$ is defined by

$$\displaystyle \begin{aligned} \begin{array}{rcl}J_{u_0,u,f}(v) & :=&\displaystyle v(t,x)- u_0(x) +i \int_0^t a_p(s)D_xv(s,x)\, ds \\ & &\displaystyle + i \sum_{j=0}^{p-1} \int_0^t \tilde{a}_j(s,x,u(s,x))D_x^j v(s,x)\, ds - i \int_0^t f(s,x)\, ds. \end{array} \end{aligned} $$

On the other hand, v solves $J_{h_0,u,D_t h}(v)=0$ if and only if it is a solution of the linear Cauchy problem

$$\displaystyle \begin{aligned}\begin{cases} \tilde{P}_u(D) v(t,x)=D_t h(t,x) \\ v(0,x)= h_0(x) \end{cases}, \end{aligned} $$

(59)

where

$$\displaystyle \begin{aligned}\tilde{P}_u(D) = D_t+a_p(t)D_x^p + \sum_{j=0}^{p-1}\tilde{a}_j(t,x,u)D_x^j.\end{aligned}$$

Notice that $\tilde {a}_j(t,x,u)$ satisfy the same conditions as a_j(t, x, u). Hence, we can apply Theorem 5 to (59), choosing η = 1 and 𝜖 = 0. It follows that there exists $v \in X_T^{s_2}$ solution of (59) satisfying the estimate (58). This concludes the proof. □

Lemma 4

The map S defined in (57) is smooth tame.

Proof

We observe that, fixed $(u_0, h_0) \in X_T^{s_2} \times X_T^{s_2},$ the constant C_n(u) in the energy estimate (58) is bounded if u belongs to a bounded neighborhood of (u₀, h₀). Evidently, from (58) we have:

$$\displaystyle \begin{aligned}| v(t, \cdot)|{}_{n,s_2}^2 \leq C^{\prime}_n |h|{}_{n+r,s_2,T}^2 \qquad t \in [0,T]\end{aligned}$$

for some $C^{\prime }_n>0$. Similarly, from the equation $\tilde {P}_u(D)v=D_th$ we get

$$\displaystyle \begin{aligned} \begin{array}{rcl} |D_t v(t, \cdot) |{}_{n, s_2} & \leq &\displaystyle |a_p(t)D^pv(t, \cdot)|{}_{n,s_2} + \sum_{j=0}^{p-1}|\tilde{a}_j (t, \cdot, u) D_x^j v(t, \cdot) |{}_{n,s_2}+|D_t h(t, \cdot)|{}_{n,s_2} \\ & \leq&\displaystyle C (|v(t, \cdot)|{}_{n+p, s_2}+ |h|{}_{n,s_2,T}) \end{array} \end{aligned} $$

for some C > 0. Hence

$$\displaystyle \begin{aligned}|S(u,h)|{}_{n,s_2,T} = \sup_{t \in [0,T]} (|v|{}_{n, s_2} + |D_t v(t, \cdot)|{}_{n, s_2} )\leq C_n |h|{}_{n+r',s_2,T} \leq C_n |(u,h)|{}_{n+r',s_2,T} \end{aligned}$$

for some C_n > 0 r′≥ 2δ(p − 1) + p. Then S is tame.

We now prove that DS is also a tame map. For $(u,h), (u_1, h_1) \in X_T^{s_2} \times X_T^{s_2}$ we have

$$\displaystyle \begin{aligned}DS(u,h)(u_1,h_1)= \lim_{\varepsilon \to 0} \frac{S(u+\varepsilon u_1, h+\varepsilon h_1)-S(u,h)}{\varepsilon} =\lim_{\varepsilon \to 0} \frac{v_\varepsilon-v}{\varepsilon}, \end{aligned}$$

where v is a solution of (59) and v_ε is the solution of

$$\displaystyle \begin{aligned}\begin{cases} \tilde{P}_{u+\varepsilon u_1}(D)v_\varepsilon = D_t (h+\varepsilon h_1) \\ v_\varepsilon (0,x)= h_0(x)+\varepsilon h_1(0,x) \end{cases}.\end{aligned}$$

A direct computation shows that the function $w_\varepsilon = \frac {v_\varepsilon - v}{\varepsilon }$ solves the Cauchy problem

$$\displaystyle \begin{aligned} \begin{cases} \tilde{P}_{u+\varepsilon u_1}w_\varepsilon =f_\varepsilon \\ w_\varepsilon(0,x)= h_1(0,x) \end{cases},\end{aligned} $$

where

$$\displaystyle \begin{aligned} f_\varepsilon = D_t h_1- \sum_{j=0}^{p-1} \frac{\tilde{a}_j(t,x, u+\varepsilon u_1)-\tilde{a}_j(t,x,u)}{\varepsilon}D_x^j v.\end{aligned} $$

We have the following: to prove that DS is tame it is sufficient to show that w_ε tends to some w₁ in $X_T^{s_2}$ for ε → 0. Indeed, this would imply that w₁ solves the Cauchy problem

$$\displaystyle \begin{aligned}\begin{cases} \tilde{P}_u(D)w_1 = f_1 \\ w_1(0,x)= h_1(0,x) \end{cases}\end{aligned}$$

where

$$\displaystyle \begin{aligned} f_1:= \lim_{\varepsilon \to 0} f_\varepsilon = D_t h_1 - \sum_{j=0}^{p-1} \partial_w \tilde{a}_j (t,x,u)u_1 D_x^j v\end{aligned} $$

and so that w₁ will satisfy an energy estimate of the form

$$\displaystyle \begin{aligned} | w_1(t,\cdot)|{}_{n,s_2}^2 \leq C_n(u) \left( | h_1(0, \cdot) |{}_{n+r,s_2}^2 + \int_0^t | f_1 (\tau, \cdot)|{}^2_{n+r,s_2}\, d\tau \right),\end{aligned} $$

(60)

which would give, by the expression of f₁,

$$\displaystyle \begin{aligned}| w_1(t,\cdot)|{}_{n,s_2} \leq C^{\prime}_n(u) (|h_1|{}_{n+r',s_2,T} + |h|{}_{n+r',s_2,T}), \qquad r'\geq 2r+p-1\end{aligned}$$

for (u, h) in a neighborhood of (u₀, h₀) and (u₁, h₁) in a neighborhood of some fixed $(\tilde {u}_1, \tilde {h}_1) \in X_T^{s_2} \times X_T^{s_2}.$ Moreover, D_tw₁ would satisfy a similar estimate, and so w₁ is tame.

Let us so prove that w_𝜖 converges in $X^{s_2}_T$ for 𝜖 → 0. Let ε₁ > 0 and ε₂ > 0 and consider the corresponding functions $w_{\varepsilon _1}$ and $w_{\varepsilon _2}$ which solve the Cauchy problems

$$\displaystyle \begin{aligned}\begin{cases} \tilde{P}_{u+\varepsilon_j u_1}(D)w_{\varepsilon_j} =f_{\varepsilon_j} \\ w_{\varepsilon_j} (0,x)= h_1(0,x) \end{cases}, \qquad j=1,2 .\end{aligned}$$

Then, it is immediate to see that $w_{\varepsilon _1}- w_{\varepsilon _2}$ is a solution of

$$\displaystyle \begin{aligned}\begin{cases} \tilde{P}_{u+\varepsilon_1 u_1}(D)(w_{\varepsilon_1}-w_{\varepsilon_2}) =f_{\varepsilon_1}-f_{\varepsilon_2} + \displaystyle\sum_{j=0}^{p-1}(\tilde{a}_j(t,x,u+\varepsilon_2 u_1)- \tilde{a}_j (t,x,u+\varepsilon_1 u_1))D_x^j w_{\varepsilon_2} \\ (w_{\varepsilon_1}-w_{\varepsilon_2}) (0,x)= 0. \end{cases}\end{aligned}$$

Then by the estimate (60) and the mean value theorem, we get

$$\displaystyle \begin{aligned} \begin{array}{rcl}| w_{\varepsilon_1}-w_{\varepsilon_2}|{}_{n,s_2} & \leq&\displaystyle C_n (u+\varepsilon_1 u_1) \left( \sup_{t \in [0,T]}| f_{\varepsilon_1}-f_{\varepsilon_2}|{}_{n+r, s_2} \right. \\ & &\displaystyle + \left.\sum_{j=0}^{p-1}\sup_{t \in [0,T]} |\partial_w \tilde a_j(t,x,u_{1,2}) (\varepsilon_1-\varepsilon_2)u_1 D_x^j w_{\varepsilon_1} |{}_{n+r,s_2} \right)\end{array} \end{aligned} $$

for some constant C_n(u + ε₁u₁) > 0 and for some u_1,2 between u + ε₁u₁ and u + ε₂u₁. Moreover, since $H_{n+r,s_2}$ is an algebra, then

$$\displaystyle \begin{aligned} \begin{array}{rcl}| \partial_w a_j(t,x,u_{1,2}) (\varepsilon_1-\varepsilon_2)& &\displaystyle u_1 D_x^j w_{\varepsilon_2} |{}_{n+r,s_2} \\ & \leq&\displaystyle | \partial_w a_j(t,x,u_{1,2})|{}_{n+r,s_2} |\varepsilon_1-\varepsilon_2| |u_1|{}_{n+r,s_2} | w_{\varepsilon_2} |{}_{n+r+j,s_2}.\end{array} \end{aligned} $$

Then, $| w_{\varepsilon _1}-w_{\varepsilon _2}|{ }_{n,s_2}$ tends to 0 when ε₁ → ε₂ → 0 if (u, h) is in a neighoborhood of (u₀, h₀) and (u₁, h₁) is in a neighborhood of some fixed $(\tilde {u}_1, \tilde {h}_1) \in X_T^{s_2} \times X_T^{s_2}.$ This shows that there exists a Cauchy sequence ε_j tending to 0 such that the corresponding function $w_{\varepsilon _j}$ converges in $X_T^{s_2}$ and this implies that DS is tame.

Using the previous results we can prove by induction on m that

$$\displaystyle \begin{aligned}D^m S(u,h)(u_1,h_1) \cdots (u_m, h_m)=w^m\end{aligned}$$

is a solution of the Cauchy problem

$$\displaystyle \begin{aligned}\begin{cases} \tilde{P}_u (D) w^m =f^m \\ w^m(0,x)= 0 \end{cases}\end{aligned}$$

with

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle f^m:= -\sum_{j=0}^{p-1}\partial_w \tilde{a}_j(t,x, u) u_m D_x^j w^{m-1}- \sum_{j=0}^{p-1}\partial_w^2 \tilde{a}_j (t,x,u) u_{m-1}u_m D_x^j w^{m-2} \\ & &\displaystyle \qquad - \cdots -\sum_{j=0}^{p-1}\partial_w^m \tilde{a}_j (t,x,u) u_1 \cdots u_{m-1}u_m D_x^j w^{0}, \end{array} \end{aligned} $$

w₀ := v, and satisfies, in a neighborhood of (u, h), (u₁, h₁), …(u_m, h_m) the estimate

$$\displaystyle \begin{aligned}|w^m|{}_{n,s_2,T} \leq C_n \sum_{j=0}^{m-1}|h_j|{}_{n+r(m),s_2,T}\end{aligned}$$

for some C_n > 0 and some $r(m) \in \mathbb N,$ where h₀ := h. The proof follows readily the argument in the proof of Lemma 3.3 in [1]. We leave the details to the reader. □

Proof of Theorem 1

We prove now the existence of a solution of the semilinear Cauchy problem (1) that is of the equation Ju = 0. We recall that Ju = 0 if and only if

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} u(t,x)& =&\displaystyle u_0(x) -i \int_0^t a_p(s)D_x^p u(s,x)\, ds \\ & -&\displaystyle i \sum_{j=0}^{p-1} \int_0^t a_j(s,x,u(s,x)) D_x^j u(s,x)\, ds + i \int_0^t f(s,x)\, ds. \end{array} \end{aligned} $$

(61)

By a linear approximation in t we get u(t, x) = w(t, x) + o(t) for t → 0 where

$$\displaystyle \begin{aligned}w(t,x)= u_0(x) -it \left( a_p(0)D_x^p u_0(x) + \sum_{j=0}^{p-1}a_j(0,x,u_0(x) ) D_x^j u_0(x) -f(0,x)\right).\end{aligned}$$

We also observe that, by the definition of J and w, we have:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \partial_t (Jw(t,x)) & =&\displaystyle \partial_t w +ia_p(t)D_x^p w +i \sum_{j=0}^{p-1}a_j(t,x,w)D_x^j w -if(t,x) \\ & =&\displaystyle i(a_p(t)-a_p(0)) D_x^pu_0+i \sum_{j=0}^{p-1}\left( a_j(t,x,w)-a_j(0,x,u_0)\right) D_x^j u_0 \\ & &\displaystyle +ta_p(t) D_x^p \left[ a_p(0) D_x^pu_0 + \sum_{j=0}^{p-1} a_j(0,x,u_0)D_x^j u_0 -f(0,x) \right] \\ & &\displaystyle + \sum_{j=0}^{p-1}t a_j(t,x,w) D_x^j \left[ a_p(0) D_x^p u_0 + \sum_{k=0}^{p-1}a_k(0,x,u_0) D_x^k u_0-f(0,x) \right] \\ & &\displaystyle + i (f(0,x)-f(t,x)). \end{array} \end{aligned} $$

From this it follows that

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & &\displaystyle |\partial_t Jw(t, \cdot )|{}_{n,s_2} \leq \sup_{t \in [0,T]}|a_p(t)-a_p(0)| \cdot | u_0|{}_{n+p,s_2} \\ & &\displaystyle \qquad + \sum_{j=0}^{p-1} | [a_j(t,x,w) -a_j(0,x,u_0)]D_x^j u_0 |{}_{n,s_2} + |f(0,x) - f(t,x)|{}_{n,s_2} \\ & &\displaystyle \qquad \qquad + t \sup_{t \in [0,T]}|a_p(t)| \cdot \left| a_p(0)D_x^p u_0 +\sum_{k=0}^{p-1}a_k(0,x,u_0)D_x^k u_0 -f(0,x) \right|{}_{n+p,s_2} \\ & &\displaystyle \qquad \qquad \qquad + t \sum_{j=0}^{p-1} \left| a_j(t,x,w) D_x^j \left[ a_p(0) D_x^p u_0 + \sum_{k=0}^{p-1}a_k(0,x,u_0) D_x^k u_0-f(0,x) \right] \right|{}_{n,s_2}. \end{array} \end{aligned} $$

(62)

Taking w in a sufficiently small neighborhood of u₀ and applying the mean value theorem to the right-hand side of (62) we obtain

$$\displaystyle \begin{aligned} |\partial_t Jw(t, \cdot) |{}_{n,s_2} \leq Ct \end{aligned} $$

(63)

for a suitable constant C = C(n, s₂, a_p, …, a₀, u₀, f). Now, fixed ε > 0 we define

$$\displaystyle \begin{aligned}\phi_\varepsilon (t,x)= \int_0^t \rho \left( \frac{s}{\varepsilon} \right) (\partial_t J w)(s,x)\, ds,\end{aligned}$$

where $\rho \in C^\infty (\mathbb R)$ such that 0 ≤ ρ ≤ 1 and ρ(s) = 0 for |s|≤ 1 and ρ(s) = 1 for |s|≥ 2. Notice that ϕ_ε = 0 for 0 ≤ t ≤ ε. Let U and V be neighborhoods of w and Jw respectively such that J : U → V is a bijection. We have that

$$\displaystyle \begin{aligned} \begin{array}{rcl} | Jw -\phi_\varepsilon|{}_{n,s_2} & =&\displaystyle \left| \int_0^t \left( 1- \rho\left( \frac{s}{\varepsilon}\right) \right) (\partial_t J w) (s, \cdot )\, ds \right|{}_{n,s_2} \\ & \leq &\displaystyle \int_0^{2\varepsilon} \left| \left( 1- \rho\left( \frac{s}{\varepsilon}\right) \right) (\partial_t J w) (s, \cdot ) \right|{}_{n,s_2}\, ds \\ & \leq &\displaystyle C \int_0^{2\varepsilon} s \, ds \leq 2C \varepsilon^2, \end{array} \end{aligned} $$

where C is the same constant appearing in (63). Similarly we obtain that

$$\displaystyle \begin{aligned}| \partial_t (Jw - \phi_\varepsilon)|{}_{n,s_2} \leq 2C \varepsilon.\end{aligned}$$

Hence, taking 0 < ε < 1 we conclude that

$$\displaystyle \begin{aligned}| Jw-\phi_\varepsilon |{}_{n,s_2,T} \leq 2 C\varepsilon.\end{aligned}$$

If we choose ε sufficiently small, we have that ϕ_ε ∈ V . Then there exists u ∈ U such that Ju = ϕ_ε. In particular we have Ju = 0 for 0 ≤ t ≤ ε, that is u is a solution in $X_{\varepsilon }^{s_2}$ of the Cauchy problem. The uniqueness of the solution comes from standard arguments, cf. [1]. □