Introduction

Our goal is to establish the stability of traveling waves to the following advection-diffusion equation with porous medium diffusion,

$$\begin{aligned} \begin{aligned} u_{t}+\left( u^{m} \right) _x=\left( u^{m} \right) _{xx}, \end{aligned} \end{aligned}$$
(1)

where \(m>0\), \(u=u(x,t)\) and the initial condition

$$\begin{aligned} u(x,0)=u_{0}(x)\rightarrow u_{\pm } \ \text {as} \ x\rightarrow \pm \infty . \end{aligned}$$
(2)

The Eq. (1) is the special case of following equation with convex nonlinearity,

$$\begin{aligned} u_t+ f(u)_x=\mu u_{xx}, \end{aligned}$$
(3)

where a constant \(\mu >0\) and a smooth function \(f^{''}(u)>0\). Il’in and Oleinik studied the stability of shock profiles to (3) based on the maximum principle [4] and spectral analysis studied by Sattinger in [19].

Moreover, the Eq. (1) reduces to Burger’s equation, when \(m=2\) (advection term) and \(m=1\) (diffusion term). Mickens and Oyedeji [17] studied traveling wave solutions to Burgers containing square root and non-diffusion Fisher,

$$\begin{aligned} \begin{aligned}&u_t+a_1\sqrt{u}u_x=D_1u_{xx},\\&u_t+a_2\sqrt{u}u_x=\lambda _1\sqrt{u}-\lambda _2u, \end{aligned} \end{aligned}$$
(4)

where \(a_1>0, a_2>0, D_1>0, \lambda _1>0, \lambda _2>0\).

Moreover, other researches containing the square root term were studied by Buckmire et. al [1], Jordan [7], and Mickens [15, 16]. This current paper, we are concerned with the stability of traveling waves of (1) with porous medium diffusion. Since there is assumption of singularity, we use the weighted energy estimates to deal with the stability of traveling waves of (1) under small perturbation and arbitrary wave amplitude as in [6, 8]. The energy method was also used to establish the stability of traveling waves to coupled Burgers equation as in [5],

$$\begin{aligned} \begin{aligned}&u_t+\left( \frac{1}{2}u^2+\frac{1}{2}b^2 \right) _x=\mu u_{xx},\\&b_t+(ub)_x=\nu b_{xx}, \end{aligned} \end{aligned}$$
(5)

where the small coefficient was not required.

A similar system to (5) was studied by Li and Wang [12], as shown the following equation

$$\begin{aligned} \begin{aligned}&u_t-(uv)_x=D u_{xx},\\&v_t+(\varepsilon v^2-u)_x=\varepsilon v_{xx}, \end{aligned} \end{aligned}$$
(6)

where this Eq. (6) was derived from Keller-Segel model, so-called chemotaxis model and the coefficients \(\varepsilon\) was assumed small enough. This research was contrast with the research in [5] where the smallness of coefficients was not required. We refer to [9, 10, 13] for other references employing the elementary energy method to study the stability problem.

The main issue of this paper are nonlinear diffusion and singular term when \(u_+=0\). We employ the weighted function to handle the singular term as studied in [6]. Moreover, the nonlinear diffusion has been studied in [3] for the traveling wave problem to chemotaxis model with \(u_+>0\).

We organize this paper as follows. We first present the existence of traveling waves to the transformed nonlinear advection-diffusion Eq. (1) in the section “Transformation of the Original Equation ”by the ansatz traveling waves, derive the appropriate perturbations, and the stability of traveling waves of (1). Moreover, we establish the a priori estimate by using the weighted energy estimates in the section“ Weighted Energy Estimate”.

Notation 1

Throughout this paper, we present the usual notation of Sobolev space \(H^r(\mathbb {R})\) where the norms are defined as \(\Vert f\Vert _r:=\sum \nolimits _{k=0}^{r}\Vert \partial _x^kf\Vert\) and \(\Vert f\Vert :=\Vert f\Vert _{L^2(\mathbb {R})}\). Moreover, we present the notation \(H^r_w(\mathbb {R})\) to represent the weighted Sobolev space where the norms are given as \(\Vert f\Vert _{r,w}:=\sum \limits _{k=0}^{r}\Vert \sqrt{w(x)}\partial _x^kf\Vert\) and \(\Vert f\Vert _w:=\Vert f\Vert _{L^2_w(\mathbb {R})}\).

Transformation of the Original Equation

The traveling wave solution of (1) can be stated in the form

$$\begin{aligned} \begin{aligned} u(x,t)=U(z),\;\;z=x-st, \end{aligned} \end{aligned}$$
(7)

satisfying

$$\begin{aligned} \begin{aligned} -sU_z+\left( U^{m} \right) _z=\left( U^{m} \right) _{zz}, \end{aligned} \end{aligned}$$
(8)

with the boundary conditions

$$\begin{aligned} \begin{aligned} U(z)\rightarrow u_\pm \;\;\text {as}\;\;z\rightarrow \pm \infty . \end{aligned} \end{aligned}$$
(9)

Now, integrating (8) with respect to z,

$$\begin{aligned} \begin{aligned} \left( U^{m} \right) _z=-sU+U^{m}+G, \end{aligned} \end{aligned}$$
(10)

where \(G=su_\pm -u_\pm ^{m}\). By using the fact that \(U_z\rightarrow 0\) as \(z\rightarrow \pm \infty\), one has Rankine-Hugoniot

$$\begin{aligned} \begin{aligned} s(u_+-u_-)=u_+^{m}-u_-^{m}. \end{aligned} \end{aligned}$$
(11)

Then by (11) with \(u_+=0\), yields

$$\begin{aligned} \begin{aligned} s=u_-^{m-1}>0. \end{aligned} \end{aligned}$$
(12)

We further, present the following proposition to deal with the existence of traveling waves

Proposition 1

Let \(u_{\pm }\) satisfy (11) and \(u_+=0\). Then there exists a monotone traveling waves \(U(x-st)\) to (8), which is unique up to a translation and satisfies \(U_z<0\), where s is given by (12). Moreover, U monotonicity behavior at \(z=\pm \infty\) with rates

$$\begin{aligned}U-u_\pm \sim e^{\lambda _\pm z},\end{aligned}$$

where \(\lambda _-=\frac{m-1}{m}\) and \(\lambda _+=1\).

Proof

It follows from (10), one has

$$\begin{aligned} \begin{aligned} U_z = \frac{U^{1-m}}{m}\cdot (-sU+U^m+G):=M(U). \end{aligned} \end{aligned}$$

We differentiate M(U) with respect to U, substitute \(G=su_\pm -u_\pm ^m\) and the points \(u_\pm\) into the results, then one yields

$$\begin{aligned} \begin{aligned} \lambda := \frac{dM(U)}{dU}=\frac{m-su_\pm ^{1-m}}{m}. \end{aligned} \end{aligned}$$

By substituting the wave speed s in (12) into \(\lambda\), and using the fact \(u_+=0\), then at the points \(u_\pm\), one can derive

$$\begin{aligned} \begin{aligned} \lambda _- = \frac{m-1}{m},\;\;\lambda _+=1, \end{aligned} \end{aligned}$$

which completes the proof. \(\square\)

Remark 1

Since we consider \(u_+=0\). Then we construct the following weighted function to handle the singularity in energy estimates

$$\begin{aligned} \begin{aligned} w(z):=1+e^{\eta z},\;\;\text {where}\;\;\eta =\frac{m-1}{m}>0\;\;\text {for}\;\; z\in \mathbb {R} \end{aligned} \end{aligned}$$

which gives

$$\begin{aligned} \begin{aligned} C_1w(z)\le \frac{1}{U(z)}\le C_2 w(z), \end{aligned} \end{aligned}$$

where \(C_2>C_1>0\).

By referring to [11, 14], let us denote the following perturbation of zero mass of \(\phi\) as

$$\begin{aligned} \phi _{0}(z)=\int _{-\infty }^{z}(u_{0}-U)(y)dy. \end{aligned}$$

Then to achieve our goal of this paper for the stability of traveling waves of (1), we give the following theorems.

Theorem 1

Let \(U(x-st)\) be the traveling waves obtained in Proposition 1. Then there exists a constant \(\varepsilon _{0}>0\) such that if \(\Vert u_{0}-U \Vert _{1,w}+\Vert \phi _{0} \Vert _w\le \varepsilon _{0}\), then the Cauchy problem (1), (2) has an unique global solution u(x, t) satisfying

$$\begin{aligned} u-U\in C \left( [0,\infty );H^{1}_{w}\right) \,\cap \,L^{2}([0,\infty );H^{1}_{w}), \end{aligned}$$

and the asymptotic stability

$$\begin{aligned} \sup _{x\in \mathbb {R}}|u(x,t)-U(x-st)|\rightarrow 0\;\;\text {as}\;\;t\rightarrow +\infty . \end{aligned}$$

By changing the variables \((x,t)\rightarrow (z=x-st,t)\), then the equation (1) becomes

$$\begin{aligned} \begin{aligned} u_{t}-su_z+(u^{m})_z=(u^{m})_{zz} \end{aligned} \end{aligned}$$
(13)

By decomposing the solution u of (13) as

$$\begin{aligned} u(z,t) = U(z)+\phi _{z}(z,t), \end{aligned}$$
(14)

then, one has

$$\begin{aligned} \begin{aligned} \phi (z,t) = \int _{-\infty }^{z}(u(y,t)-U(y))dy. \end{aligned} \end{aligned}$$
(15)

Now, we substitute (14) into (13) and integrating the resuls in z, one has

$$\begin{aligned} \begin{aligned} \phi _{t} =s\phi _z-(\phi _z+U)^{m}+m\left( U^{m-1}\phi _z \right) _{z}+F \end{aligned} \end{aligned}$$
(16)

where \(F=\left( (\phi _z+U)^{m}-U^{m}-mU^{m-1}\phi _z \right) _z\) and initial data of \(\phi\)

$$\begin{aligned} \phi (z,0) = \phi _{0}(z)=\int _{-\infty }^{z}(u_{0}-U)dy \end{aligned}$$
(17)

with \(\phi _{0}(\pm \infty )=0\). We further find the solution of transformed problem (16), (17) in the space

$$\begin{aligned} X(0,T):=\left\{ \phi (z,t)\in C([0,T),H^{2}_{w}): \phi _{z}\in L^{2}((0,T);H^{2}_{w}))\right\} \end{aligned}$$

with \(0<T\le +\infty\). Let

$$\begin{aligned} N(t):=\sup _{0\le \tau \le t}\left\{ \Vert \phi (.,\tau ) \Vert _{2,w}\right\} . \end{aligned}$$

From the Sobolev inequality \(\Vert f\Vert _{L^\infty }\le \sqrt{2}\Vert f\Vert ^{\frac{1}{2}}_{L^2_w}\Vert f_x\Vert ^{\frac{1}{2}}_{L^2_w}\), it follows that

$$\begin{aligned} \sup _{\tau \in [0,t]}\{\Vert \phi (\cdot ,\tau )\Vert _{L^\infty },\Vert \phi _{z}(\cdot ,\tau )\Vert _{L^\infty }\}\le N(t). \end{aligned}$$

For (16), (17), we have the following global well-posedness.

Theorem 2

There exists a constant \(\delta _{1}>0\), such that if \(N(0)\le \delta _{1}\), then Cauchy problem (16), (17) has an unique global solution \(\phi \in X(0,+\infty )\) such that

$$\begin{aligned} \begin{aligned} \Vert \phi (.,t)\Vert _{2,w}^{2}+\int _{0}^{t}\Vert \phi _{z}(.,\tau )\Vert _{2,w}^{2}\;d\tau \le C\Vert \phi _{0}\Vert _{2,w}^{2}\le CN^2(0), \end{aligned} \end{aligned}$$
(18)

for any \(t\in [0,\infty )\) and the asymptotic stability

$$\begin{aligned} \sup _{z\in R} |\phi _{z}(z,t)|\rightarrow 0 \ \text {as} \ t\rightarrow +\infty . \end{aligned}$$
(19)

The global existence of \(\phi\) stated in Theorem 2 follows from the local existence theorem and the a priori estimates which are given below.

Proposition 2

(Local existence) For any \(\varepsilon _1>0\), there exists a positive constant T depending on \(\varepsilon _1\) such that if \(\phi \in H^2_w\) with \(N(0)\le \varepsilon _1/2\), then problem (16), (17) has an unique solution \(\phi \in X(0,T)\) satisfying \(N(t)\le 2N(0)\) for any \(0\le t\le T\).

By referring to [18], we can establish the local existence under a standard way.

Proof

We differentiate (16) in z twice that

$$\begin{aligned} \begin{aligned} \phi _{zzt}&=s\phi _{zzz}-m(\phi _z+U)^{m-1}(\phi _{zzz}+U_{zz})\\&\quad -m(m-1)(\phi _z+U)^{m-2}(\phi _{zz}+U_{z})^2+m(U^{m-1}\phi _z)_{zzz}+F_{zz}. \end{aligned} \end{aligned}$$

Then multiplying by \(\phi _{zz}/U\), integrating the results in t, and applying \(N(t)\ll 1\), one yields

$$\begin{aligned} \begin{aligned} \int \phi _{zz}^2 w+\int _0^t\int \phi ^2_{zzz}w\le C\int \phi _{0zz}^2w+CN(t)\int ^t_0\int \phi _{zz}^2w, \end{aligned} \end{aligned}$$

for some \(C>0\). By Sobolev embedding theorem, all the terms on the right hand side can be bounded by

$$\begin{aligned} \begin{aligned} CN(t)\int _0^t\int \phi _{zzz}^2w, \end{aligned} \end{aligned}$$

then one gets

$$\begin{aligned} \begin{aligned} N^2(t)+\int ^t_0\int \phi ^2_{zzz}w\le CN^2(0)+CN(t)\int ^t_0\int \phi _{zzz}^2w. \end{aligned} \end{aligned}$$

By choosing \(C=4\) such that \(N(t)\le 1/4\), one has

$$\begin{aligned} \begin{aligned} N(t)\le 2N(0), \end{aligned} \end{aligned}$$

which completes the proof of Proposition 2. \(\square\)

We further need to establish the a priori estimate.

Proposition 3

(A priori estimates) Let \(\phi \in X(0,T)\) be a solution obtained in Proposition 2 for some time \(T>0\). Then there exists a constant \(\varepsilon _2>0\), independent of T such that if \(N(t)<\varepsilon _2\), then \(\phi\) satisfies (18) for any \(0\le t\le T\).

The proof of Proposition 3 is based on the following energy estimates.

Lemma 1

Let \(\phi \in H^2_w(\mathbb {R})\) and \(\phi\) be solution of (16), (17), then there exists a constant \(C>0\) such that

$$\begin{aligned} \Vert \phi (.,t)\Vert ^{2}_{2,w}+\int _{0}^{t}\Vert \phi _{z}(.,\tau )\Vert ^{2}_{2,w}d\tau \le C\Vert \phi _{0}\Vert ^{2}_{2,w}+CN(t)\int _0^t\int \phi _{zz}^2w \end{aligned}$$
(20)

Then, we can prove the Proposition 3 through the Lemma 1.

Proof

To establish the result, it only suffices to show that the a priori estimate (18) holds. By applying the Sobolev embedding theorem, we noting that all the nonlinear terms on the right-hand side of (20) can be bounded by

$$\begin{aligned} CN(t)\int _{0}^{t}\Vert \phi _{z}(.,\tau )\Vert ^{2}_{2,w}d\tau , \end{aligned}$$

for some \(C>0\). Then it follows from Lemma 1, one has

$$\begin{aligned} N^2(t)+\int _{0}^{t}\Vert \phi _{z}(.,\tau )\Vert ^{2}_{2,w}d\tau \le CN^2(0)+ CN(t)\int _{0}^{t}\Vert \phi _{z}(.,\tau )\Vert ^{2}_{2,w}d\tau , \end{aligned}$$

for \(0\le t\le T\) and some constant \(C>0\). For any \(t\in [0,T]\), we choose \(N(t)\le 1/2C\) to get

$$\begin{aligned} N^2(t)+\int _{0}^{t}\Vert \phi _{z}(.,\tau )\Vert ^{2}_{2,w}d\tau \le CN^2(0), \end{aligned}$$

which gives the result (18). \(\square\)

We are now ready to prove Theorem 2 which is the main theorem in this paper.

Proof

By having the transformation (14), Theorem 1 is a consequence of Theorem 2. The a priori estimate (18) guarantees that N(t) is small if N(0) is small enough. Thus, applying the standard extension procedure, we get the global well-posedness of (16), (17) in \(X(0,+\infty )\).

Next, we prove the convergence (19). Clearly, if \(\phi \in H^2_w\), then \(\phi \in H^2\), since \(w\ge 1\). Owing to the global estimate (18), we get

$$\begin{aligned} \int _{0}^{t}\int _{-\infty }^{\infty }\phi _z^2(z,\tau )dzd\tau \le C\Vert \phi _{0}\Vert _{2,w}^{2}\le CN^2(0). \end{aligned}$$
(21)

In view of the first equation of (16), one has

$$\begin{aligned}\begin{aligned} \frac{d}{dt}\int _{-\infty }^{\infty }\phi _z^2(z,t)dz&=-2 \int _{-\infty }^{\infty }\phi _t\phi _{zz}dz\\&=-2 \int _{-\infty }^{\infty }\phi _{zz}\left( s\phi _z-(\phi _z+U)^{m}+m(U^{m-1}\phi _z)_z+F \right) . \end{aligned}\end{aligned}$$

Then, by employing the binomial theorem and Young’s inequality, one yields

$$\begin{aligned}\begin{aligned} \frac{d}{dt}\int _{-\infty }^{\infty }\phi _z^2(z,t)dz&\le 2 \int _{-\infty }^{\infty }\phi _{zz}\left( s\phi _z+u_-^{m}({m}!)^2\phi _z^2\sum _{l=0}^{{m}}\frac{1}{l!}\left( \frac{1}{u_-} \right) ^l+\phi _{zz} \right) \\&\quad +2\int _{-\infty }^{\infty }\phi _{zz}\left( m(U^{m-1}\phi _z)_z+2C\phi _{z}\phi _{zz}+C\phi _{zz} \right) \\&=2 \int _{-\infty }^{\infty }\phi _{zz}\left( s\phi _z+u_-^{m}({m}!)^2\phi _z^2e^{1/u_-}+\phi _{zz} \right) \\&\quad +2\int _{-\infty }^{\infty }\phi _{zz}\left( m(U^{m-1}\phi _z)_z+2C\phi _{z}\phi _{zz}+C\phi _{zz} \right) \\&\le C\int _{-\infty }^{\infty }(\phi _{zz}^2+\phi _{z}^2). \end{aligned}\end{aligned}$$

It then follows from the global estimate (18) that

$$\begin{aligned} \begin{aligned} \int _0^\infty \left| \frac{d}{dt}\int _{-\infty }^{\infty }\phi _z^2(z,t)dz\right| \le C\int _0^\infty \int _{-\infty }^{\infty }(\phi _{zz}^2+\phi _{z}^2)\le C\Vert \phi _{0}\Vert ^{2}_{2,w}\le CN^2(0).\end{aligned}\end{aligned}$$
(22)

From (21) and (22), we get

$$\begin{aligned} \int _{-\infty }^{\infty }\phi _z^2(z,t)dz\rightarrow 0 \ \text {as} \ t\rightarrow +\infty .\end{aligned}$$

By Cauchy-Schwarz inequality, we further have

$$\begin{aligned} \begin{aligned} \phi _{z}^{2}(z,t)&=2\int _{-\infty }^{z}\phi _{z}\phi _{zz}(y,t)dy\\&\le 2\left( \int _{-\infty }^{+\infty }\phi _{z}^{2}(y,t)dy \right) ^{\frac{1}{2}}\left( \int _{-\infty }^{+\infty }\phi _{zz}^{2}(y,t)dy \right) ^{\frac{1}{2}}\\&\le C\left( \int _{-\infty }^{+\infty }\phi _{z}^{2}(y,t)dy \right) ^{\frac{1}{2}}\rightarrow 0 \ \text {as} \ t\rightarrow +\infty \end{aligned} \end{aligned}$$

Hence (19) is proved. \(\square\)

Weighted Energy Estimate

Now, we are concerned with the a priori estimates for solution \(\phi\) of (16), (17), and hence prove Proposition 3. To avoid the singular term, we apply the weighted function \(1/U(z)\le Cw\) for all \(z\in \mathbb {R}\) as studied in [6, 8].

\(L^2\)-estimate of \(\phi\)

Lemma 2

Under the assumptions of Lemma 1, if \(N(t)\ll 1\), then

$$\begin{aligned} \begin{aligned} \Vert \phi (.,t)\Vert ^{2}_w+\int _{0}^{t}\Vert \phi _{z}(.,\tau )\Vert ^{2}_wd\tau \le C\Vert \phi _{0}\Vert ^{2}_w+CN(t)\int _0^t\int \phi ^2_{zz}w \end{aligned} \end{aligned}$$
(23)

Proof

We multiply the Eq. (16) by \(\phi /U\) and integrate the result in z to get

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\int \frac{\phi ^2}{U}+m\int U^{m-2}\phi _z^2\\&=\int \frac{s\phi \phi _z}{U}-\int \frac{(\phi _z+U)^{m}\phi }{U}-\int m U^{m-1}\phi \phi _z\left( \frac{1}{U} \right) _z+\int \frac{F\phi }{U}. \end{aligned} \end{aligned}$$
(24)

We estimate the term \((\phi _z+U)^m\) in (24) that

$$\begin{aligned} \begin{aligned} (\phi _z+U)^{m}\le (\phi _z+u_-)^{m}=u_-^{m}\left( \frac{\phi _z}{u_-}+1 \right) ^{m}=\sum _{l=0}^{m}u_-^{m}\frac{P^{m}_l}{l!}\left( \frac{\phi _z}{u_-} \right) ^l, \end{aligned} \end{aligned}$$
(25)

where \(P^{m}_l=\frac{m!}{(m-l)!}\).

Since \(N(t)\ll 1\), which implies \(\left\| \phi _z(\cdot ,t) \right\| _{L^\infty }\le 1\), then (25) becomes

$$\begin{aligned} \begin{aligned} (\phi _z+U)^{m}\le u_-^{m}(m!)^2\phi _z^2\sum _{l=0}^{m}\frac{1}{l!}\left( \frac{1}{u_-} \right) ^l=u_-^{m}(m!)^2\phi _z^2e^{1/u_-}\le C\phi _z^2. \end{aligned} \end{aligned}$$
(26)

Substituting (26) into (24), one has

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\int \frac{\phi ^2}{U}+m\int U^{m-2}\phi _z^2\\&\le C\int \frac{\phi \phi _z^2}{U}-\int \frac{\phi ^2}{2}\left[ \left( \frac{s}{U} \right) _z-mU^{m-1}\left( \frac{1}{U}\right) _{zz} \right] . \end{aligned} \end{aligned}$$
(27)

It follows from (10), and the fact \(u_+=0\), noting that

$$\begin{aligned} \begin{aligned} \left( \frac{s}{U} \right) _z-mU^{m-1}\left( \frac{1}{U}\right) _{zz}&=\left[ \left( \frac{s}{U} \right) +mU^{m-1}\frac{U_z}{U^2} \right] _z\\&=\left[ \frac{s}{U}+\frac{-sU+U^{m}+su_+-u_+^{m}}{U^2} \right] _z\\&=\left[ \frac{U^{m}+su_+-u_+^{m}}{U^2} \right] _z\\&=(m-2)U^{m-2}\frac{U_z}{U}. \end{aligned} \end{aligned}$$
(28)

By Young’s inequality, \(\left\| \phi (\cdot ,t) \right\| _{L^\infty }\le N(t)\ll 1\), and (26), it holds

$$\begin{aligned} \begin{aligned} \left| \int \frac{F\phi }{U} \right| \le CN(t)\int \left( \frac{|\phi _z|^2+|\phi _{zz}|^2}{U} \right) . \end{aligned} \end{aligned}$$
(29)

Substituting (28), (29) into (27), then applying \(1/U(z)\le Cw\) for all \(z\in \mathbb {R}\), one has

$$\begin{aligned} \begin{aligned} \frac{1}{2}\frac{d}{dt}\int \phi ^2 w+(1-CN(t))\int \phi ^2_zw\le CN(t)\int \phi ^2_{zz}w. \end{aligned} \end{aligned}$$
(30)

We further integrate (30) in t and using the fact \(N(t)\ll 1\), then the proof of Lemma 2 is finished. \(\square\)

\(H^1\)-estimate of \(\phi\)

Lemma 3

Under the assumptions of Lemma 1, if \(N(t)\ll 1\), then

$$\begin{aligned} \begin{aligned} \Vert \phi (.,t)\Vert ^{2}_{1,w}+\int _{0}^{t}\Vert \phi _{z}(.,\tau )\Vert ^{2}_{1,w}d\tau \le C\Vert \phi _{0}\Vert ^{2}_{1,w}. \end{aligned} \end{aligned}$$
(31)

Proof

We differentiate (16) in z to get

$$\begin{aligned} \begin{aligned} \phi _{zt}=s\phi _{zz}-m(\phi _z+U)^{m-1}(\phi _{zz}+U_z)+m(U^{m-1}\phi _z)_{zz}+F_z. \end{aligned} \end{aligned}$$
(32)

Now, multiplying (31) by \(\phi _z/U\), one has

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\int \frac{\phi _z^2}{U}+m\int U^{m-2}\phi _{zz}^2\\&=\int \frac{s\phi _z\phi _{zz}}{U}-\int mU^{m-1}\phi _{z}\phi _{zz}\left( \frac{1}{U} \right) _z-\int \frac{m(\phi _z+U)^{m-1}(\phi _{zz}+U_z)\phi _z}{U}\\&\quad +\int m(m-1)U^{m-1}\left( \frac{1}{U} \right) _z\phi _z\phi _{zz}+\int m(m-1)U^{m}\left( \phi _z\left( \frac{1}{U} \right) _z \right) ^2\\&\quad +\int |F|\left( |\phi _z|+\frac{|\phi _{zz}|}{U} \right) . \end{aligned} \end{aligned}$$
(33)

By the similar way in \(L^2\) estimate, we can also estimate the term \((\phi _z+U)^{m-1}\) in (33) which consists of two conditions: \((\phi _z+U)^{m-1}\le (\phi _z+u_-)^{m-1}\) for \(m\ge 1\) and \((\phi _z+U)^{m-1}\le (\phi _z+Cw(z))^{m-1}\) for \(0<m<1\), where w(z) is defined in Remark 1. Then, it follows from (16), \(0<U(z)\le u_-\) and (26), we can derive

$$\begin{aligned} \begin{aligned} |F|\le C \left( |\phi _{z}||\phi _{zz}|+|\phi _z|^2 \right) . \end{aligned} \end{aligned}$$
(34)

By Young’s inequality and \(\left\| \phi (\cdot ,t) \right\| _{L^\infty }\le N(t)\), one has

$$\begin{aligned} \begin{aligned} \int |F|\left( |\phi _z|+\frac{|\phi _{zz}|}{U} \right) \le CN(t)\int \left( \frac{|\phi _{z}|^2+|\phi _{zz}|^2}{U} \right) . \end{aligned} \end{aligned}$$
(35)

Substituting (34), (35) into (33) to get

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\int \frac{\phi _z^2}{U}+(1-CN(t))\int \frac{\phi _{zz}^2}{U}\\&\le C\int \frac{\phi _z^4+U_z\phi _z^3}{U}-\int \frac{\phi _z^2}{2}\left[ \left( \frac{s}{U} \right) -m U^{m-1}\left( \frac{1}{U}\right) _z \right] _z\\&\quad +\int \frac{CN(t)}{U}(|\phi _{z}|^2+|\phi _{zz}|^2). \end{aligned} \end{aligned}$$
(36)

Substituting (28) into (36), applying Young’s inequality, \(\left\| \phi _z \right\| _{L^\infty }\ll N(t)\), \(1/U(z)\le Cw\) for all \(z\in \mathbb {R}\), and the results with (23), one has

$$\begin{aligned} \begin{aligned} \frac{1}{2}\frac{d}{dt}\int \phi _z^2 w+(1-CN(t))\int \phi ^2_{zz}w\le CN(t)\int \phi _z^2w. \end{aligned} \end{aligned}$$
(37)

Integrating (37) with respect to t and combining the results with Lemma 2, one has

$$\begin{aligned} \begin{aligned} \int \phi _z^2 w+(1-CN(t))\int _0^t\int \phi ^2_{zz}w\le C\int \phi _{0z}^2w. \end{aligned} \end{aligned}$$
(38)

Using the fact \(N(t)\ll 1\), then the proof of Lemma 3 is finished. \(\square\)

\(H^2\)-estimate of \(\phi\)

Lemma 4

Under the assumptions of Lemma 1, if \(N(t)\ll 1\), then

$$\begin{aligned} \begin{aligned} \Vert \phi (.,t)\Vert ^{2}_{2,w}+\int _{0}^{t}\Vert \phi _{z}(.,\tau )\Vert ^{2}_{2,w}d\tau \le C\Vert \phi _{0}\Vert ^{2}_{2,w} \end{aligned} \end{aligned}$$
(39)

Proof

We differentiate (32) in z to get

$$\begin{aligned} \begin{aligned} \phi _{zzt}&=s\phi _{zzz}-m(\phi _z+U)^{m-1}(\phi _{zzz}+U_{zz})\\&\quad -m(m-1)(\phi _z+U)^{m-2}(\phi _{zz}+U_{z})^2+m(U^{m-1}\phi _z)_{zzz}+F_{zz}. \end{aligned} \end{aligned}$$
(40)

Now, multiplying (31) by \(\phi _{zz}/U\), one has

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\int \frac{\phi _{zz}^2}{U}+m\int U^{m-2}\phi _{zzz}^2\\&=\int \frac{s\phi _{zz}\phi _{zzz}}{U}-\int mU^{m-1}\phi _{zz}\phi _{zzz}\left( \frac{1}{U} \right) _z-\int \frac{m(\phi _z+U)^{m-1}(\phi _{zzz}+U_{zz})\phi _{zz}}{U}\\&\quad +\int m(m-1)U^{m-1}\left( \frac{1}{U} \right) _z\phi _{zz}\phi _{zzz}+\int m(m-1)U^{m}\left( \phi _{zz}\left( \frac{1}{U} \right) _z \right) ^2\\&\quad +\int m^2(m-1)U^{m}\left( \frac{1}{U} \right) _z^2\frac{\phi _z\phi _{zz}\phi _{zzz}}{U}+\int m^2(m-1)U^{m-1}U_z\left( \frac{1}{U} \right) _z^2\frac{\phi _z\phi _{zz}^2}{U}\\&\quad +\int m(m-1)U^{m}\left( \frac{1}{U} \right) _{zz}\frac{\phi _z\phi _{zz}\phi _{zzz}}{U^2}+\int m(m-1)U^{m-1}\left( \frac{1}{U} \right) _{zz}\left( \frac{1}{U} \right) _{z}\frac{\phi _z\phi _{zz}^2}{U}\\&\quad +\int \frac{F_{zz}\phi _{zz}}{U}-\int \frac{m(m-1)(\phi _z+U)^{m-2}(\phi _{zz}+U_z)^2\phi _{zz}}{U}. \end{aligned} \end{aligned}$$
(41)

We employ the similar technique as in \(L^2\) and \(H^1\) estimates to establish the estimate of the term \((\phi _z+U)^{m-2}\). Noting that

$$\begin{aligned} \begin{aligned} \int \frac{F_{zz}\phi _{zz}}{U}=-\int \frac{F_z\phi _{zzz}}{U}+\int \frac{F_z\phi _{zz}U_z}{U^2}, \end{aligned} \end{aligned}$$
(42)

and

$$\begin{aligned} \begin{aligned} F_z&=m((U+\phi _z)^{m-1}-U^{m-1})\phi _{zzz}+m(m-1)(U+\phi _z)^{m-2}\phi _{zz}^2\\&\quad +m(m-1)U_z^2 \left( (U+\phi _z)^{m-2}-U^{m-2}-(m-2)U^{m-3}\phi _z \right) \\&\quad +mU_{zz}((U+\phi _z)^{m-1}-U^{m-1}-(m-1)U^{m-2}\phi _z)\\&\quad +2m(m-1)U_z((U+\phi _z)^{m-2}-U^{m-2})\phi _{zz}. \end{aligned} \end{aligned}$$
(43)

By Young’s inequality, \(\left\| \phi _z(\cdot ,t) \right\| _{L^\infty }\le N(t)\), (43), and (34), then (42) becomes

$$\begin{aligned} \begin{aligned} \left| \int \frac{F_{zz}\phi _{zz}}{U} \right| \le CN(t)\int \left( \frac{|\phi _{zz}|^2+|\phi _{zzz}|^2}{U} \right) . \end{aligned} \end{aligned}$$
(44)

Substituting (42)–(44) into (41), one has

$$\begin{aligned} \begin{aligned}&\frac{1}{2}\frac{d}{dt}\int \frac{\phi _{zz}^2}{U}+(1-CN(t))\int \frac{\phi _{zzz}^2}{U}\\&\le -\int \frac{\phi _{zz}^2}{2}\left[ \left( \frac{s}{U} \right) -m U^{m-1}\left( \frac{1}{U}\right) _z \right] _z+\int CN(t)\frac{|\phi _{zz}|^2}{U}. \end{aligned} \end{aligned}$$
(45)

Substituting (28) into (45), applying Young’s inequality, \(\left\| \phi _z \right\| _{L^\infty }\ll N(t)\), \(1/U(z)\le Cw\) for all \(z\in \mathbb {R}\), the results in (23) and (31), and integrating with respect to t, then one has

$$\begin{aligned} \begin{aligned} \int \phi _{zz}^2 w+(1-CN(t))\int _0^t\int \phi ^2_{zzz}w\le C\int \phi _{0zz}^2w. \end{aligned} \end{aligned}$$
(46)

Using the fact \(N(t)\ll 1\), then the proof of Lemma 4 is finished. \(\square\)