Asymptotic expansion of solutions to the periodic problem for a non-linear Sobolev-type equation

In this paper we study the long-time asymptotic behaviour of solutions to the periodic problem for the non-linear Sobolev-type equation

$$\begin{equation} {\begin{array}{c} \partial _{t}(v-\Delta v)-\alpha \Delta v =\lambda v+\beta \operatorname{div}(|\nabla v|\nabla v), \qquad x\in T^{n}, \quad t>0, \\ v(0,x)=\widetilde{v}(x), \qquad x\in T^{n}, \end{array}} \end{equation} \tag{ 1 }$$

where $T^{n}$ is the $n$-dimensional torus (that is, the cube $\Omega =([-\pi ,\pi ])^{n}$ with natural identification of the appropriate faces), $\alpha ,\sigma >0$ and $\lambda >-\alpha$, $\beta \in \mathbb {R}$. We consider spatially $2\pi$-periodic solutions $v(t,x)$ of (1) with $2\pi$-periodic initial data $\widetilde{v}(x)$.

Note that the evolution of the mean value $\int _{\Omega }v(t,x)\,dx$ can easily be calculated as

$$\begin{equation*} \int _{\Omega }v(t,x)\,dx=Me^{\lambda t}, \end{equation*}$$

where $M\equiv \int _{\Omega }\widetilde{v}(x)\,dx$ is the mean value of the initial perturbation $\widetilde{v}(x)$. Changing the variables by

$$\begin{equation*} v(t,x)=e^{\lambda t}M+e^{\lambda t}u(t,x), \end{equation*}$$

we get the periodic problem

$$\begin{equation} {\begin{array}{c} \partial _{t}(u-\Delta u)-\gamma \Delta u=\beta e^{\sigma \lambda t} \operatorname{div}(|\nabla u|^{\sigma }\nabla u), \qquad x\in T^{n}, \quad t>0, \\ u(0,x)=\varphi (x), \qquad x\in T^{n}, \end{array}} \end{equation} \tag{ 2 }$$

whose initial data $\varphi (x)=\widetilde{v}(x)-M$ have zero mean value, and with $\gamma =\alpha +\lambda$. By the assumption $\lambda >-\alpha$ we have $\gamma >0$. Thus, in what follows we study the periodic problem (2) with $\gamma >0$, $\beta \in \mathbb {R}$ and

$$\begin{equation} \int _{\Omega }\varphi (x)\,dx=0. \end{equation} \tag{ 3 }$$

Sobolev-type equations [1] describe various physical processes and have been studied in many papers. The monograph [2] gives a comprehensive survey of the modern theory of linear and non-linear Sobolev-type equations along with a derivation of linear higher-order Sobolev-type modelling equations arising in the theory of plasma and the description of quasistationary processes in electromagnetic continua. The long-time asymptotics of solutions to the Cauchy problem for non-linear Sobolev-type equations with various types of non-linearities was studied in the book [3] and the survey [4].

Many publications are devoted to the study of space-periodic problems. For example, asymptotic stability of stationary periodic solutions of Fisher's equation was proved in [5], the periodic problem for the Hopf equation was studied in [6], and the long-time asymptotic behaviour of solutions to the periodic problem for the Burgers equation was obtained in [7]. Some decay estimates for the solutions to the Korteweg–de Vries–Burgers equation with periodic initial data having zero mean value have been found in [8]. It was also proved there that the asymptotic behaviour is determined by higher-order harmonics. The papers [9], [10] contain a proof of the existence and uniqueness of global-in-time solutions to the periodic problem for the Landau–Ginzburg equation as well as decay estimates with respect to time in various norms. Explicit periodic solutions to the Landau–Ginzburg equation have been found in [11]. The global existence and decay estimates with respect to time for solutions to the periodic problem for the Kuramoto–Sivashinsky equation were obtained in [12]. For the Benjamin–Bona–Mahoney–Peregrin–Burgers equation, the existence of global solutions and energy decay estimates with respect to time have been established in [13]. The periodic problem for the Boussinesq equation was studied in [14]. The long-time behaviour of solutions to the periodic problem for systems of conservation laws was studied in [15], [16]. In [17] the authors consider the issues of existence and blow-up of solutions to the periodic problem for the Camassa–Holm equation. Asymptotic formulae for the solutions to the periodic problem for various non-linear equations have been obtained in [18].

As far as we know, the long-time behaviour of solutions to the periodic problem for the non-linear Sobolev-type equation (1) has not been treated before. Our aim is to prove the asymptotic formulae for the solutions to the periodic problem for the non-linear Sobolev-type equation (1).

We introduce some notation. Let

$$\begin{equation*} \widehat{\varphi }_{k} =\frac{1}{(2\pi )^{n}}\int _{\Omega }e^{-i(k\cdot x)}\varphi (x)\,dx \end{equation*}$$

be the Fourier coefficients of a $2\pi$-periodic function $\varphi (x)$. As usual, by $\mathbf {W}_{p}^{2}(\Omega )$ we denote the Sobolev space with norm

$$\begin{equation*} \Vert \varphi \Vert _{\mathbf {W}_{p}^{2}(\Omega )}\equiv \Vert (1-\Delta )\varphi \Vert _{\mathbf {L}^{p}(\Omega )}, \end{equation*}$$

where $1\leqslant p\leqslant \infty$, and we write $\mathbf {H}^{2}(\Omega )\equiv \mathbf {W}_{2}^{2}(\Omega )$. We denote by $\mathbf {C}(\mathbf {I};\mathbf {B})$ the space of continuous functions from a temporal interval $\mathbf {I}$ to a Banach space $\mathbf {B}$. We denote various constants by the same symbol $C$.

Using the Duhamel principle, we rewrite the periodic problem (2) in the form of an integral equation

$$\begin{equation} u(t)=\mathcal {G}(t)\varphi +\beta \int _{0}^{t}e^{\sigma \lambda \tau }\mathcal {G}(t-\tau )(1-\Delta )^{-1} \operatorname{div}\bigl (|\nabla u|^{\sigma }\nabla u(\tau )\bigr )\,d\tau , \end{equation} \tag{ 4 }$$

where the Green operator $\mathcal {G}(t)$ can be formally represented as a Fourier series

$$\begin{equation} \mathcal {G}(t)\psi =\sum _{k\in \mathbb {Z}^{n}}\widehat{\psi }_{k}e^{i(x\cdot k)-L_{k}t} \end{equation} \tag{ 5 }$$

with $\smash[t]{L_{k}=\frac{\gamma |k|^{2}}{1+|k|^{2}}}$. By a global-in-time solution of the periodic problem (2) we mean a solution $u\in \mathbf {C}([0,\infty );\mathbf {W}_{p}^{2}(\Omega ))$ of the corresponding integral equation (4).

We now state the main result of this paper.

Theorem 1. Let $\sigma >0$ and $\lambda <\frac{\gamma }{2}$. Suppose that $\varphi \in \mathbf {W}_{p}^{2}(\Omega )$, $p>n$ for $n\geqslant 1$, and the norm $\Vert \varphi \Vert _{\mathbf {W}_{p}^{2}}$ is sufficiently small. Then there is a unique global-in-time solution $u\in \mathbf {C}([0,\infty );\mathbf {W}_{p}^{2}(\Omega ))$ to the periodic problem (2). It satisfies the estimate

$$\begin{equation*} \Vert u(t)\Vert _{\mathbf {L}^{\infty }}\leqslant Ce^{-\frac{\gamma }{2}t} \qquad \forall \,t>0. \end{equation*}$$

For equations (2) with $\beta >0$ we can relax the requirement that the initial data should be small. For simplicity we consider only the case when $n=1$.

Theorem 2. Let $\sigma >0$, $\beta >0$ and $n=1$. Suppose that $\varphi \in \mathbf {H}^{2}(\Omega )$. Then there is a unique solution $u\in \mathbf {C}([0,\infty );\mathbf {H}^{2}(\Omega ))$ to the periodic problem (2). Moreover, it satisfies the decay estimate

$$\begin{equation*} \Vert u(t)\Vert _{\mathbf {L}^{\infty }}\leqslant Ce^{-\frac{\gamma }{2}t} \qquad \forall \,t>0. \end{equation*}$$

In the following theorem we obtain an asymptotic expansion of the solution. We put $L_{k}\equiv L_{|k|}=\frac{\gamma |k|^{2}}{1+|k|^{2}}$.

Theorem 3. Let $\sigma >1$ and $\lambda \leqslant \frac{\gamma }{2}(1-\frac{1}{\sigma })$. Suppose that $\varphi \in \mathbf {W}_{p}^{2}(\Omega )$, $p>n$ for $n\geqslant 1$, and the norm $\Vert \varphi \Vert _{\mathbf {W}_{p}^{2}}$ is sufficiently small. Then there are numbers $\widehat{a}_{k}$ (the Fourier coefficients of the function $a$ defined in (16) below) such that the following asymptotic formula holds:

$$\begin{equation*} u(t)=\sum _{|k|\leqslant N}e^{i(x\cdot k)-L_{k}t}\widehat{a}_k +O(e^{-tL_{N+1}}) \end{equation*}$$

as $t\rightarrow \infty$ uniformly with respect to $x\in \Omega$, where $N\geqslant 0$.

In the next section we obtain some preliminary estimates for the Green operator of the periodic problem for the linear Sobolev-type equation and prove the global existence of solutions to the periodic problem for the non-linear equation (1). In the three subsequent sections we prove Theorems 1–3 respectively.

Consider the periodic problem for the linear Sobolev-type equation

$$\begin{equation} {\begin{array}{c} \partial _{t}(u-\Delta u)-\gamma \Delta u=f, \qquad x\in \Omega , \quad t>0, \\ u(0,x)=\varphi (x), \qquad x\in \Omega , \end{array}} \end{equation} \tag{ 6 }$$

where the right-hand side $f(t,x)$ and the initial perturbation $\varphi (x)$ are $2\pi$-periodic functions of the spatial variable $x$. Using Fourier series, we can formally write the Green operator $\mathcal {G}(t)$ in the form (5). Therefore the solution to the periodic problem for the linear Sobolev-type equation (6) can be written by means of the Duhamel formula as

$$\begin{equation} u(t)=\mathcal {G}(t)\varphi +\int _{0}^{t}\mathcal {G}(t-\tau )(1-\Delta )^{-1}f(\tau )\,d\tau . \end{equation} \tag{ 7 }$$

We first estimate the solution to the linear periodic problem (6) in terms of the Lebesgue norms in $\mathbf {L}^{p}$, $1\leqslant p\leqslant \infty$. We introduce two projectors

$$\begin{equation*} \mathcal {P}_{N}u=\sum _{|k|\leqslant N}\widehat{u}_k(t)e^{i(k\cdot x)}, \qquad \mathcal {R}_{N}u=\sum _{|k|\geqslant N}\widehat{u}_k(t)e^{i(k\cdot x)} \end{equation*}$$

for $N\geqslant 0$. Then $\mathcal {P}_{N}u+\mathcal {R}_{N+1}u=u$ for every $N\geqslant 0$.

Lemma 1. Suppose that $\varphi \in \mathbf {L}^{p}$ and $f(t)\in \mathbf {L}^{p}$ for $1\leqslant p\leqslant \infty$. Then for $N\geqslant 0$ and $1\leqslant p\leqslant \infty$ we have

$$\begin{equation} \Vert \mathcal {R}_{N}{\mathcal {G}}(t)\varphi \Vert _{\mathbf {L}^{p}} \leqslant Ce^{-tL_{N}}\Vert \varphi \Vert _{\mathbf {L}^{p}} \end{equation} \tag{ 8 }$$

for all $t>0$,

$$\begin{equation} \biggl \Vert \int _{0}^{t}\mathcal {R}_{N}{\mathcal {G}}(t-t_1) f(t_1)\,dt_1\biggr \Vert _{\mathbf {L}^{p}} \leqslant Ce^{-tL_{N}}\int _{0}^{t}e^{t_1(L_{N}-\Lambda )}\,dt_1\cdot \sup _{\tau >0}e^{\Lambda \tau }\Vert \mathcal {R}_{N}f(\tau )\Vert _{\mathbf {L}^{p}} \end{equation} \tag{ 9 }$$

for all $t>0$ if $\Lambda \in \mathbb {R}$, and

$$\begin{equation} \biggl \Vert \int _{t}^{\infty }\mathcal {P}_{N}{\mathcal {G}}(t-t_1) f(t_1)\,dt_1\biggr \Vert _{\mathbf {L}^{p}} \leqslant Ce^{-\Lambda t}\sup _{\tau >0} e^{\Lambda \tau }\Vert \mathcal {P}_{N}f(\tau )\Vert _{\mathbf {L}^{p}} \end{equation} \tag{ 10 }$$

for all $t>0$ if $\Lambda >L_{N}$.

Proof. We write the representation

$$\begin{equation} \begin{split} e^{-tL_{k}} & =e^{-\gamma t}\biggl (e^{\gamma t\langle k\rangle ^{-2}} -\sum _{j=0}^{n}\frac{\gamma ^{j}t^{j}}{j!}\langle k\rangle ^{-2j}\biggr ) +e^{-\gamma t}\sum _{j=0}^{n}\frac{\gamma ^{j}t^{j}}{j!}\langle k\rangle ^{-2j} \\ & =e^{-\gamma t}\frac{1}{n!}\int _{0}^{\gamma t\langle k\rangle ^{-2}} e^{z}(\gamma t\langle k\rangle ^{-2}-z)^{n}\,dz +e^{-\gamma t}\sum _{j=0}^{n}\frac{\gamma ^{j}t^{j}}{j!}\langle k\rangle ^{-2j} \end{split} \end{equation} \tag{ 11 }$$

for $|k|\geqslant 0$, where $\langle k\rangle =\sqrt{1+|k|^{2}}$. Note that the remainder term

$$\begin{equation*} Q_{k}(t)=e^{-\gamma t}\frac{1}{n!}\int _{0}^{\gamma t\langle k\rangle ^{-2}} e^{z}(\gamma t\langle k\rangle ^{-2}-z)^{n}\,dz \end{equation*}$$

satisfies the estimate

$$\begin{equation*} |Q_{k}(t)|\leqslant Ct^{n+1}e^{-tL_{k}}\langle k\rangle ^{-2n-2} \end{equation*}$$

for $|k|\geqslant 0$. We put $\mathcal {B}^{0}=\mathbf {1}$ and define the operators

$$\begin{equation*} \mathcal {B}^{j}\varphi =\int _{\Omega }B_{j}(x-y)\varphi (y)\,dy \end{equation*}$$

with kernels

$$\begin{equation*} B_{j}(x)=\frac{1}{(2\pi )^{n}}\sum _{k\in \mathbb {Z}^{n}} e^{i(x\cdot k)}\langle k\rangle ^{-2j} \end{equation*}$$

for $j\geqslant 1$, and the remainder operator $\mathcal {Q}_{N}$,

$$\begin{equation*} \mathcal {Q}_{N}\varphi =\int _{\Omega }\widetilde{Q}_{N}(t,x-y)\varphi (y)\,dy, \end{equation*}$$

with kernel

$$\begin{equation*} \widetilde{Q}_{N}(t,x)=\frac{1}{(2\pi )^{n}}\sum _{|k|\geqslant N} e^{i(x\cdot k)}Q_{k}(t) \end{equation*}$$

such that by (11) we have the representation

$$\begin{equation} \begin{split} \mathcal {R}_{N}{\mathcal {G}}(t)\varphi & =\sum _{|k|\geqslant N}\widehat{\varphi }_{k}e^{i(x\cdot k)-L_{k}t} \\ & =\sum _{|k|\geqslant N}e^{i(x\cdot k)}Q_{k}(t)\widehat{\varphi }_{k} +e^{-\gamma t}\sum _{j=0}^{n}\frac{\gamma ^{j}t^{j}}{j!}\sum _{|k|\geqslant N} e^{i(x\cdot k)}\langle k\rangle ^{-2j}\widehat{\varphi }_{k} \\ & =\mathcal {Q}_{N}\mathcal {R}_{N}\varphi +e^{-\gamma t}\sum _{j=0}^{n}\frac{\gamma ^{j}t^{j}}{j!} \mathcal {B}^{j}\mathcal {R}_{N}\varphi . \end{split} \end{equation} \tag{ 12 }$$

Then $B_{j}(x)\in \mathbf {C}(\Omega \setminus \lbrace 0\rbrace )$ and we have

$$\begin{equation*} |B_{j}(x)|\leqslant C+C|x|^{2j-n} \end{equation*}$$

for all $j\geqslant 1$ and $x\in \Omega \setminus \lbrace 0\rbrace$ (see [19], Theorem 2.17). We also have

$$\begin{equation*} |\widetilde{Q}_{N}(t,x)|\leqslant \sum _{|k|\geqslant N}|Q_{k}(t)| \leqslant Ct^{n+1}e^{-L_{N}t}. \end{equation*}$$

Therefore

$$\begin{equation} \begin{gathered} \Vert \mathcal {B}^{j}\varphi \Vert _{\mathbf {L}^{p}} \leqslant C\Vert \varphi \Vert _{\mathbf {L}^{p}} \qquad \forall \,j\geqslant 0, \end{gathered} \end{equation} \tag{ 13 }$$

$$\begin{equation} \begin{gathered} \Vert \mathcal {Q}_{N}(t)\varphi \Vert _{\mathbf {L}^{p}} \leqslant Ct^{n+1}e^{-L_{N}t}\Vert \varphi \Vert _{\mathbf {L}^{p}}, \qquad 1\leqslant p\leqslant \infty . \end{gathered} \end{equation} \tag{ 14 }$$

Thus, by (13) and (14), we obtain from (12) that

$$\begin{equation*} \begin{aligned} \Vert \mathcal {R}_{N}{\mathcal {G}}(t)\varphi \Vert _{\mathbf {L}^{p}} & \leqslant Ct^{n+1}e^{-L_{N}t}\Vert \mathcal {R}_{N}\varphi \Vert _{\mathbf {L}^{p}} +Ct^{n}e^{-\gamma t}\Vert \mathcal {R}_{N}\varphi \Vert _{\mathbf {L}^{p}} \\ &\leqslant Ct^{n+1}e^{-L_{N}t}\Vert \mathcal {R}_{N}\varphi \Vert _{\mathbf {L}^{p}}. \end{aligned} \end{equation*}$$

Since $L_{N+1}=\gamma -\frac{\gamma }{1+(N+1)^{2}}>L_{N} =\gamma -\frac{\gamma }{1+N^{2}}$ and

$$\begin{equation*} \mathcal {R}_{N}{\mathcal {G}}(t)\varphi =\sum _{|k|=N}\widehat{\varphi }_{k}e^{i(x\cdot k)-L_{k}t} +\mathcal {R}_{N+1}{\mathcal {G}}(t)\varphi , \end{equation*}$$

we get estimate (8) in the lemma:

$$\begin{equation} \Vert \mathcal {R}_{N}{\mathcal {G}}(t)\varphi \Vert _{\mathbf {L}^{p}} \leqslant e^{-L_{N}t}\Vert \mathcal {R}_{N}\varphi \Vert _{\mathbf {L}^{1}} +\Vert \mathcal {R}_{N+1}{\mathcal {G}}(t)\varphi \Vert _{\mathbf {L}^{p}} \leqslant Ce^{-L_{N}t}\Vert \mathcal {R}_{N}\varphi \Vert _{\mathbf {L}^{p}} \end{equation} \tag{ 15 }$$

for all $t>0$. We similarly have

$$\begin{equation} \begin{split} \Vert \mathcal {P}_{N}\mathcal {G}(t)\varphi \Vert _{\mathbf {L}^{p}} & =\biggl \Vert \sum _{|k|\leqslant N}\widehat{\varphi }_{k}(t) e^{i(x\cdot k)-L_{k}t}\biggr \Vert _{\mathbf {L}^{p}} \\ & \leqslant Ce^{-L_{N}t}\sum _{|k|\leqslant N}|\widehat{\varphi }_{k}(t)| \leqslant C_{N}e^{-L_{N}t}\Vert \mathcal {P}_{N}\varphi \Vert _{\mathbf {L}^{p}} \end{split} \end{equation} \tag{ 16 }$$

for all $t<0$. Using (15) and (16), we obtain

$$\begin{equation*} \begin{aligned} \biggl \Vert \int _{0}^{t}\mathcal {R}_{N}{\mathcal {G}}(t-\tau ) f(\tau )\,d\tau \biggr \Vert _{\mathbf {L}^{p}} &\leqslant C\int _{0}^{t}\Vert \mathcal {R}_{N}{\mathcal {G}}(t-\tau ) f(\tau )\Vert _{\mathbf {L}^{p}}\,d\tau \\ & \leqslant C\int _{0}^{t}e^{-(t-\tau )L_{N}}\Vert \mathcal {R}_{N} f(\tau )\Vert _{\mathbf {L}^{p}}\,d\tau \\ & \leqslant Ce^{-tL_{N}}\int _{0}^{t}e^{\tau (L_{N}-\Lambda )}\,d\tau \cdot \sup _{\tau >0}e^{\Lambda \tau }\Vert \mathcal {R}_{N}f(\tau )\Vert _{\mathbf {L}^{p}} \end{aligned} \end{equation*}$$

for all $t>0$ if $\Lambda \in \mathbb {R}$. We similarly get

$$\begin{equation*} \begin{aligned} \biggl \Vert \int _{t}^{\infty }\mathcal {P}_{N}{\mathcal {G}}(t-\tau ) f(\tau )\,d\tau \biggr \Vert _{\mathbf {L}^{p}} &\leqslant C_{N}\int _{t}^{\infty }e^{(\tau -t)L_{N}}\Vert \mathcal {P}_{N} f(\tau )\Vert _{\mathbf {L}^{p}}\,d\tau \\ & \leqslant C_{N}e^{-tL_{N}}\int _{t}^{\infty }e^{-\tau (\Lambda -L_{N})}\,d\tau \cdot \sup _{\tau >0}e^{\Lambda \tau }\Vert \mathcal {P}_{N}f(\tau )\Vert _{\mathbf {L}^{p}} \\ & \leqslant C_{N}e^{-\Lambda t}\sup _{\tau >0}e^{\Lambda \tau }\Vert \mathcal {P}_{N} f(\tau )\Vert _{\mathbf {L}^{p}} \end{aligned} \end{equation*}$$

for all $t>0$ if $\Lambda >L_{N}$. □

In the following lemma we establish the local-in-time existence of solutions of the periodic problem (2).

Lemma 2. Let $\sigma >0$ and $\lambda \leqslant \frac{\gamma }{2}$. Suppose that $\varphi \in \mathbf {W}_{p}^{2}(\Omega )$ and $p>n$ for $n\geqslant 1$. Then for some time $T>0$ there is a unique solution $u\in \mathbf {C}([0,T];\mathbf {W}_{p}^{2}(\Omega ))$ of the periodic problem (2).

Proof. We put

$$\begin{equation*} \mathcal {N}(u(\tau ))=\beta e^{\sigma \lambda \tau } \operatorname{div}(|\nabla u(\tau )|^{\sigma }\nabla u(\tau )) \end{equation*}$$

and consider the integral equation (4):

$$\begin{equation*} u(t)=\mathcal {G}(t)\varphi +\int _{0}^{t}\mathcal {G}(t-\tau )(1-\Delta )^{-1}\mathcal {N}(u(\tau ))\,d\tau . \end{equation*}$$

By (3) we have $\varphi =\mathcal {R}_{1}\varphi$ and $u=\mathcal {R}_{1}u$. Consider the ball

$$\begin{equation*} \mathbf {X}_{T}=\Bigl \lbrace u\in \mathbf {C}([0,T];\mathbf {W}_{p}^{2}(\Omega )) :\sup _{t\in [0,T]}e^{\frac{\gamma }{2}t}\Vert u(t)\Vert _{\mathbf {W}_{p}^{2}} \leqslant \rho \Bigr \rbrace . \end{equation*}$$

For all $v\in \mathbf {X}_{T}$ we define the map

$$\begin{equation*} \mathcal {M}(v)=\mathcal {G}(t)\varphi +\int _{0}^{t}\mathcal {G}(t-\tau )(1-\Delta )^{-1}\mathcal {N}(v(\tau ))\,d\tau . \end{equation*}$$

We claim that $\mathcal {M}(v)\in \mathbf {X}_{T}$. Indeed, the Sobolev embedding theorem (see [20]) yields that $\Vert v\Vert _{\mathbf {L}^{\infty }}\leqslant C\Vert v\Vert _{\mathbf {W}_{p}^{1}}$ if $p>n$ for $n\geqslant 1$. Hence,

$$\begin{equation*} \begin{aligned} & \Vert (1-\Delta )^{-1}\mathcal {N}(v(\tau ))\Vert _{\mathbf {W}_{p}^{2}} \leqslant C\Vert \mathcal {N}(v)\Vert _{\mathbf {L}^{p}} \\ & \qquad \qquad \leqslant Ce^{\sigma \lambda \tau }\Vert |\nabla v|^{\sigma }\Delta v(\tau )\Vert _{\mathbf {L}^{p}}\leqslant Ce^{\sigma \lambda \tau } \Vert \Delta v(\tau )\Vert _{\mathbf {L}^{p}} \Vert \nabla v(\tau )\Vert _{\mathbf {W}_{p}^{1}}^{\sigma } \\ & \qquad \qquad \leqslant Ce^{\sigma \lambda \tau -\frac{\gamma }{2}(1+\sigma )\tau } \Bigl (\sup _{t\in [0,T]}e^{\frac{\gamma }{2}t} \Vert v(t)\Vert _{\mathbf {W}_{p}^{2}}\Bigr )^{1+\sigma } \leqslant C\rho ^{1+\sigma }e^{\sigma \lambda \tau -\frac{\gamma }{2}(1+\sigma )\tau }. \end{aligned} \end{equation*}$$

In view of estimate (8) in Lemma 1 we have

$$\begin{equation*} \begin{aligned} \Vert \mathcal {R}_{1}{\mathcal {G}}\varphi \Vert _{\mathbf {W}_{p}^{2}} &\leqslant \Vert \mathcal {R}_{1}{\mathcal {G}}\varphi \Vert _{\mathbf {L}^{p}} +\Vert \mathcal {R}_{1}{\mathcal {G}}\Delta \varphi \Vert _{\mathbf {L}^{p}} \\ & \leqslant Ce^{-\frac{\gamma }{2}t}\Vert \varphi \Vert _{\mathbf {L}^{p}} +Ce^{-\frac{\gamma }{2}t}\Vert \Delta \varphi \Vert _{\mathbf {L}^{p}} \leqslant Ce^{-\frac{\gamma }{2}t}\Vert \varphi \Vert _{\mathbf {W}_{p}^{2}} \end{aligned} \end{equation*}$$

for all $t>0$. Using estimate (9) in Lemma 1 for $\Lambda =-\sigma \lambda +\frac{\gamma }{2}(1+\sigma )$, we get

$$\begin{equation*} \begin{aligned} & \biggl \Vert \int _{0}^{t}\mathcal {R}_{1}{\mathcal {G}}(t-\tau )(1-\Delta )^{-1} \mathcal {N}(v(\tau ))\,d\tau \biggr \Vert _{\mathbf {W}_{p}^{2}} \\ & \qquad \qquad \qquad \leqslant Ce^{-\frac{\gamma }{2}t}\int _{0}^{t} e^{\tau (\frac{\gamma }{2}-\Lambda )}\,d\tau \cdot \sup _{\tau \in [0,T]}e^{\Lambda \tau }\Vert (1-\Delta )^{-1} \mathcal {N}(v(\tau ))\Vert _{\mathbf {W}_{p}^{2}} \\ & \qquad \qquad \qquad \leqslant C\rho ^{1+\sigma }e^{-\frac{\gamma }{2}t}\int _{0}^{t} e^{-\sigma \tau (\frac{\gamma }{2}-\lambda )}\,d\tau \leqslant CTe^{-\frac{\gamma }{2}t}\rho ^{1+\sigma } \end{aligned} \end{equation*}$$

for all $t\in [0,T]$ since $\lambda \leqslant \frac{\gamma }{2}$. It follows that $\mathcal {M}(v)\in \mathbf {X}_{T}$ for a suitable choice of $\rho$ and $T$.

We now claim that $\mathcal {M}$ is a contraction. Indeed, as above, we have

$$\begin{equation*} \begin{aligned} \Vert (1-\Delta )^{-1}(\mathcal {N}(v)-\mathcal {N}(w))\Vert _{\mathbf {W}_{p}^{2}} &\leqslant Ce^{\sigma \lambda \tau }\Vert \Delta (v-w)\Vert _{\mathbf {L}^{p}} \bigl (\Vert \nabla v\Vert _{\mathbf {W}_{p}^{1}}^{\sigma } +\Vert \nabla w\Vert _{\mathbf {W}_{p}^{1}}^{\sigma }\bigr ) \\ & \leqslant C\rho ^{\sigma }e^{\sigma \lambda \tau -\frac{\gamma }{2}(1+\sigma )\tau } \sup _{t\in [0,T]}e^{\frac{\gamma }{2}t}\Vert v(t)-w(t)\Vert _{\mathbf {W}_{p}^{2}}, \end{aligned} \end{equation*}$$

whence we deduce from estimate (9) in Lemma 1 that

$$\begin{equation*} \begin{aligned} & e^{\frac{\gamma }{2}t}\biggl \Vert \int _{0}^{t}\mathcal {R}_{1}{\mathcal {G}} (t-\tau )(1-\Delta )^{-1}(\mathcal {N}(v(\tau )) -\mathcal {N}(w(\tau )))\,d\tau \biggr \Vert _{\mathbf {W}_{p}^{2}} \\ &\qquad \qquad \leqslant C\int _{0}^{t}e^{\tau (\frac{\gamma }{2}-\Lambda )}\,d\tau \cdot \sup _{\tau \in [0,T]}e^{\Lambda \tau }\Vert (1-\Delta )^{-1}(\mathcal {N}(v(\tau )) -\mathcal {N}(w(\tau )))\Vert _{\mathbf {L}^{p}} \\ & \qquad \qquad \leqslant CT\rho ^{\sigma }\sup _{t\in [0,T]}e^{\frac{\gamma }{2}t} \Vert v(t)-w(t)\Vert _{\mathbf {W}_{p}^{2}} \leqslant \frac{1}{2}\sup _{t\in [0,T]}e^{\frac{\gamma }{2}t} \Vert v(t)-w(t)\Vert _{\mathbf {W}_{p}^{2}} \end{aligned} \end{equation*}$$

for all $t\in [0,T]$ if $T>0$ is chosen sufficiently small. Thus $\mathcal {M}$ is a contraction in the space $\mathbf {X}_{T}$ and, therefore, there is a unique solution $u\in \mathbf {C}([0,T];\mathbf {W}_{p}^{2}(\Omega ))$ of the periodic problem (2). □

Consider the ball

$$\begin{equation*} \mathbf {X}_{\infty }=\Bigl \lbrace u\in \mathbf {C}([0,\infty ); \mathbf {W}_{p}^{2}(\Omega )):\sup _{t>0}e^{\frac{\gamma }{2}t} \Vert u(t)\Vert _{\mathbf {W}_{p}^{2}}\leqslant \rho \Bigr \rbrace \end{equation*}$$

of a small radius $\rho >0$. As in the proof of Lemma 2, we can see that $\mathcal {M}$ is a contraction in the space $\mathbf {X}_{\infty }$. Hence there is a unique global-in-time solution $u\in \mathbf {C}([0,\infty );\mathbf {W}_{p}^{2}(\Omega ))$ of the periodic problem (2). This solution satisfies the estimate

$$\begin{equation*} \Vert u(t)\Vert _{\mathbf {W}_{p}^{2}}\leqslant \rho e^{-\frac{\gamma }{2}t} \end{equation*}$$

for all $t>0$. By the Sobolev embedding theorem, this yields the estimate in the theorem. □

Multiplying the one-dimensional version

$$\begin{equation*} \partial _{t}(u-\partial _{x}^{2}u)-\gamma \partial _{x}^{2}u =\beta e^{\sigma \lambda t}\partial _{x}(|u_{x}|^{\sigma }u_{x}) \end{equation*}$$

of equation (2) by $2u$ and integrating over $x\in \Omega$, we get

$$\begin{equation*} \frac{d}{dt}\bigl (\Vert u(t)\Vert _{\mathbf {L}^{2}}^{2} +\Vert u_{x}(t)\Vert _{\mathbf {L}^{2}}^{2}\bigr ) =-2\gamma \Vert u_{x}(t)\Vert _{\mathbf {L}^{2}}^{2}-2\beta e^{\sigma \lambda t}\Vert u_{x}(t)\Vert _{\mathbf {L}^{\sigma +2}}^{\sigma +2}. \end{equation*}$$

In a similar vein, multiplying (2) by $2u_{xx}$ and integrating over $x\in \Omega$, we get

$$\begin{equation} \frac{d}{dt}\bigl (\Vert u_{x}(t)\Vert _{\mathbf {L}^{2}}^{2} +\Vert u_{xx}(t)\Vert _{\mathbf {L}^{2}}^{2}\bigr ) =-2\gamma \Vert u_{xx}(t)\Vert _{\mathbf {L}^{2}}^{2}-2\beta (\sigma +1) e^{\sigma \lambda t}\Vert |u_{x}|^{\sigma }u_{xx}^{2}\Vert _{\mathbf {L}^{1}}. \end{equation} \tag{ 17 }$$

In particular, the norm $\Vert u(t)\Vert _{\mathbf {H}^{2}}$ is bounded for all $t>0$. Using estimate (11) and the standard extension of solutions, we find that there is a global-in-time solution $u\in \mathbf {C}([0,\infty );\mathbf {H}^{2}(\Omega ))$ of the periodic problem (2).

By Parseval's identity we have

$$\begin{equation*} \Vert u(t)\Vert _{\mathbf {L}^{2}}^{2}=\sum _{|k|\geqslant 1}|\widehat{u}_k(t)|^{2} \leqslant \sum _{|k|\geqslant 1}\langle k\rangle ^{2}|\widehat{u}_k(t)|^{2} =\Vert u_{x}(t)\Vert _{\mathbf {L}^{2}}^{2}. \end{equation*}$$

We similarly have $\Vert u_{x}(t)\Vert _{\mathbf {L}^{2}}\leqslant \Vert u_{xx}(t)\Vert _{\mathbf {L}^{2}}$. Hence it follows from (17) that

$$\begin{equation} \frac{d}{dt}J\leqslant -2\gamma \Vert u_{xx}(t)\Vert _{\mathbf {L}^{2}}^{2}\leqslant -\gamma J, \end{equation} \tag{ 18 }$$

where $J=\Vert u_{x}(t)\Vert _{\mathbf {L}^{2}}^{2}+\Vert u_{xx}(t)\Vert _{\mathbf {L}^{2}}^{2}$. Integrating (18) with respect to $t$, we get the estimate

$$\begin{equation*} \Vert u(t)\Vert _{\mathbf {H}^{2}}\leqslant CJ^{\frac{1}{2}}(t) \leqslant Ce^{-\frac{\gamma }{2}t} \end{equation*}$$

for all $t\geqslant 0$. Theorem 2 now follows from the Sobolev embedding theorem. □

We put

$$\begin{equation*} u_{N}(t)=\mathcal {P}_{N}u =\sum _{|k|\leqslant N}\widehat{u}_{k}(t)e^{i(k\cdot x)}, \qquad r_{N}(t)=\mathcal {R}_{N+1}u =\sum _{|k|\geqslant N+1}\widehat{u}_{k}(t)e^{i(k\cdot x)} \end{equation*}$$

for $N\geqslant 0$. Thus we get a representation $u=u_{N}+r_{N}$. It follows from the integral equation (4) that

$$\begin{equation} r_{N}(t)=\mathcal {R}_{N+1}\mathcal {G}(t)\varphi +\int _{0}^{t}\mathcal {R}_{N+1}\mathcal {G}(t-\tau ) (1-\Delta )^{-1}\mathcal {N}(u(\tau ))\,d\tau , \end{equation} \tag{ 19 }$$

where $\mathcal {N}(u(\tau )) =\beta e^{\sigma \lambda \tau }\operatorname{div} (|\nabla u(\tau )|^{\sigma }\nabla u(\tau ))$. We define the space

$$\begin{equation*} \mathbf {Y}_{N}=\lbrace \psi \in \mathbf {C}([0,\infty );\mathbf {W}_{p}^{2}(\Omega )) :\Vert \psi \Vert _{\mathbf {Y}_{N}}<\infty \rbrace \end{equation*}$$

with the norm

$$\begin{equation*} \Vert \psi \Vert _{\mathbf {Y}_{N}} =\sup _{t>0}e^{L_{N+1}t}\Vert \psi (t)\Vert _{\mathbf {W}_{p}^{2}}. \end{equation*}$$

We now use induction to prove that

$$\begin{equation} \Vert r_{N}\Vert _{\mathbf {Y}_{N}}\leqslant C_{N} \end{equation} \tag{ 20 }$$

for all $N\geqslant 0$. Since $u_{0}(t)=0$ and $r_{0}(t)=u(t)$, the estimate (20) for $N=0$ can be obtained in the same way as in the proof of Lemma 1. Suppose that (20) holds for $0\leqslant N<N_{1}$ and consider $N=N_{1}\geqslant 1$.

By estimate (8) in Lemma 1 we have

$$\begin{equation*} \Vert \mathcal {R}_{N+1}\mathcal {G}(t)\varphi \Vert _{\mathbf {Y}_{N}} =\sup _{t>0}e^{L_{N+1}t}\Vert \mathcal {R}_{N+1} \mathcal {G}(t)\varphi \Vert _{\mathbf {W}_{p}^{2}} \leqslant C\Vert \varphi \Vert _{\mathbf {W}_{p}^{2}}. \end{equation*}$$

Using estimate (9) in Lemma 1, we find that

$$\begin{equation*} \begin{aligned} & \biggl \Vert \int _{0}^{t}\mathcal {R}_{N+1}\mathcal {G}(t-\tau )(1-\Delta )^{-1} \mathcal {N}(u(\tau ))\,d\tau \biggr \Vert _{\mathbf {Y}_{N}} \\ & \qquad \qquad \qquad =\sup _{t>0}e^{L_{N+1}t}\biggl \Vert \int _{0}^{t}\mathcal {R}_{N+1}\mathcal {G} (t-\tau )(1-\Delta )^{-1}\mathcal {N}(u(\tau ))\,d\tau \biggr \Vert _{\mathbf {W}_{p}^{2}} \\ & \qquad \qquad \qquad \leqslant C\sup _{t>0}\int _{0}^{t}e^{\tau (L_{N+1}-\gamma )}\,d\tau \cdot \sup _{\tau >0}e^{\gamma \tau }\Vert (1-\Delta )^{-1}\mathcal {R}_{N+1} \mathcal {N}(u(\tau ))\Vert _{\mathbf {W}_{p}^{2}}. \end{aligned} \end{equation*}$$

By the Sobolev embedding theorem (see [20]) we have $\Vert u\Vert _{\mathbf {L}^{\infty }}\leqslant C\Vert u\Vert _{\mathbf {W}_{p}^{1}}$ for $p>n\geqslant 1$, whence

$$\begin{equation} \begin{aligned} & e^{\gamma \tau }\Vert (1-\Delta )^{-1}\mathcal {R}_{N+1} \mathcal {N}(u(\tau ))\Vert _{\mathbf {W}_{p}^{2}} \leqslant Ce^{\gamma \tau }\Vert \mathcal {N}(u(\tau ))\Vert _{\mathbf {L}^{p}} \\ & \qquad \qquad \qquad \leqslant Ce^{\sigma \lambda \tau +\gamma \tau }\Vert |\nabla u(\tau )|^{\sigma } \Delta u(\tau )\Vert _{\mathbf {L}^{p}} \leqslant Ce^{\sigma \lambda \tau +\gamma \tau }\Vert \Delta u(\tau )\Vert _{\mathbf {L}^{p}} \Vert \nabla u(\tau )\Vert _{\mathbf {W}_{p}^{1}}^{\sigma } \\ & \qquad \qquad \qquad \leqslant Ce^{\sigma \lambda \tau +\gamma \tau -\frac{\gamma }{2}(1+\sigma )\tau } \Bigl (\sup _{t>0}e^{\frac{\gamma }{2}t} \Vert u(t)\Vert _{\mathbf {W}_{p}^{2}}\Bigr )^{1+\sigma }\leqslant C_{N}, \end{aligned} \end{equation} \tag{ 21 }$$

because $\smash{\lambda \leqslant \frac{\gamma }{2}-\frac{\gamma }{2\sigma }}$. It follows that

$$\begin{equation*} \biggl \Vert \int _{0}^{t}\mathcal {R}_{N+1}\mathcal {G}(t-\tau ) (1-\Delta )^{-1}\mathcal {N}(u(\tau ))\,d\tau \biggr \Vert _{\mathbf {Y}_{N}} \leqslant C_{N}. \end{equation*}$$

Thus estimate (20) holds for all $N\geqslant 0$.

We now find the asymptotics of the solution. We put

$$\begin{equation} a=\varphi +\int _{0}^{\infty }\mathcal {G}(-\tau )(1-\Delta )^{-1} \mathcal {N}(u(\tau ))\,d\tau \end{equation} \tag{ 22 }$$

and write the representation

$$\begin{equation} u_{N+1}(t)=\mathcal {P}_{N+1}\mathcal {G}(t)a-\int _{t}^{\infty } \mathcal {P}_{N+1}\mathcal {G}(t-\tau )(1-\Delta )^{-1} \mathcal {N}(u(\tau ))\,d\tau \end{equation} \tag{ 23 }$$

for $N\geqslant 0$. By estimate (10) in Lemma 1 we have

$$\begin{equation*} \begin{aligned} & \biggl \Vert \int _{t}^{\infty }\mathcal {P}_{N+1}\mathcal {G}(t-\tau ) (1-\Delta )^{-1}\mathcal {N}(u(\tau ))\,d\tau \biggr \Vert _{\mathbf {Y}_{N}} \\ & \qquad \qquad \qquad \qquad \qquad =\sup _{t>0}e^{L_{N+1}t}\biggl \Vert \int _{t}^{\infty }\mathcal {P}_{N+1} \mathcal {G}(t-\tau )(1-\Delta )^{-1} \mathcal {N}(u(\tau ))\,d\tau \biggr \Vert _{\mathbf {W}_{p}^{2}} \\ & \qquad \qquad \qquad \qquad \qquad \leqslant C\sup _{t>0}e^{(L_{N+1}-\gamma )t} \sup _{\tau >0}e^{\gamma \tau }\Vert \mathcal {P}_{N+1}(1-\Delta )^{-1} \mathcal {N}(u(\tau ))\Vert _{\mathbf {W}_{p}^{2}}. \end{aligned} \end{equation*}$$

As in (21), we have the estimate

$$\begin{equation*} e^{\gamma \tau }\Vert \mathcal {P}_{N+1}(1-\Delta )^{-1} \mathcal {N}(u(\tau ))\Vert _{\mathbf {W}_{p}^{2}} \leqslant e^{\gamma \tau }\Vert \mathcal {N}(u(\tau ))\Vert _{\mathbf {L}^{p}}\leqslant C_{N}. \end{equation*}$$

Therefore we get

$$\begin{equation} \biggl \Vert \int _{t}^{\infty }\mathcal {P}_{N+1}\mathcal {G}(t-\tau ) (1-\Delta )^{-1}\mathcal {N}(u(\tau ))\,d\tau \biggr \Vert _{\mathbf {Y}_{N}} \leqslant C_{N}. \end{equation} \tag{ 24 }$$

Furthermore, since

$$\begin{equation*} \mathcal {P}_{N+1}\mathcal {G}(t)a =\sum _{|k|\leqslant N+1}e^{i(x\cdot k)-L_{k}t}\,\widehat{a}_k, \end{equation*}$$

we obtain the following asymptotic formula from (20), (23), (24):

$$\begin{equation*} \begin{aligned} u=u_{N+1}+r_{N+1} &=\sum _{|k|\leqslant N+1}e^{i(x\cdot k)-L_{k}t}\,\widehat{a}_k +O(e^{-tL_{N+1}}) \\ & =\sum _{|k|\leqslant N}e^{i(x\cdot k)-L_{k}t}\,\widehat{a}_k+O(e^{-tL_{N+1}}) \end{aligned} \end{equation*}$$

as $t\rightarrow \infty$ uniformly with respect to $x\in \Omega$. □