Periodic solutions of a quasilinear parabolic equation with nonlinear convection terms

Abstract

In this paper, we study a periodic quasilinear parabolic equation with nonlinear convection terms and weakly nonlinear sources. Based on the theory of the Leray-Schauder fixed point theorem, we establish the existence of periodic solutions when the domain of the solution is sufficiently small.

1 Introduction

In this paper, we consider the following periodic quasilinear parabolic equation with nonlinear convection terms and weakly nonlinear sources:

(1.1)

(1.2)

(1.3)

where Ω is a bounded domain in ${\mathbb{R}}^N$ with a smooth boundary ∂ Ω, $Q_T=\mathrm{\Omega }\times (0,T)$, and we assume that

(A1) $a_{ij}(\cdot ,\cdot ,u)=a_{ji}(\cdot ,\cdot ,u)\in C_T({\overline{Q}}_T)$ and there exist two constants $0<\lambda \le \mathrm{\Lambda }$ such that

$$\lambda {|\xi |}^2\le a_{ij}(x,t,u){\xi }_i{\xi }_j\le \mathrm{\Lambda }{|\xi |}^2,\mathrm{\forall }(x,t)\in Q_T,\xi \in {\mathbb{R}}^{+}.$$

(A2) $B(x,t,u)$ is Hölder continuous in $\overline{\mathrm{\Omega }}\times \mathbb{R}\times \mathbb{R}$, periodic in t with a period T and satisfies $B(x,t,u)u\le b_0{|u|}^{\alpha +1}$ with constants $b_0\ge 0$ and $0\le \alpha \le 1$.

(A3) $h(x,t)\in C_T({\overline{Q}}_T)\cap L^{\mathrm{\infty }}(0,T;W_0^{1,\mathrm{\infty }}(\mathrm{\Omega }))$, $h(x,t)>0$ for $\mathrm{\Omega }\times \mathbb{R}$, where $C_T({\overline{Q}}_T)$ denotes the set of functions which are continuous in $\overline{\mathrm{\Omega }}\times \mathbb{R}$ and ω-periodic with respect to t.

The existence of periodic solutions for parabolic equations has been considered by several authors; see [1–12] and the references therein. As a work related to this paper, we refer to Nakao [10], in which the author considered the following parabolic equation:

$$\frac{\partial u}{\partial t}-\mathrm{\Delta }\beta (u)=B(x,t,u)+h(x,t),$$

with Dirichlet boundary value conditions, where B, h are periodic in t with a period $\omega >0$, $\beta (u)$ satisfies ${\beta }^{\prime }(u)>0$ except for $u=0$ and $\beta (u)$ is fulfilled by ${|u|}^{m-1}u$ if $m>1$. Under the assumption that $B(x,t,u)u\le b_0|u|$, Nakao established the existence of periodic solutions by the Leray-Schauder fixed point theorem. In [12], Zhou et al. considered the quasilinear parabolic equation with nonlocal terms. Based on the theory of Leray-Schauder’s degree, the authors established the existence of nontrivial periodic solutions. In this paper, we consider the quasilinear parabolic equation (1.1) with weakly nonlinear sources and nonlinear convection terms. The convection term $b(u)\cdot \mathrm{\nabla }u$ describes an effect of convection with a velocity field $b(u)$. Under a restrictive condition that the domain is sufficiently small, we establish the existence of periodic solutions of the problem (1.1)-(1.3).

This paper is organized as follows. The definition of the generalized solution and a useful a priori estimate are presented in Section 2. Our main results will be given in Section 3.

2 Preliminaries

Our main efforts will focus on the discussion of generalized solutions since the regularity follows from a quite standard approach. Hence, we give the following definition of generalized solutions.

Definition 1 A function u is said to be a generalized solution of the problem (1.1)-(1.3) if $u\in L^2(0,T;H_0^1(\mathrm{\Omega }))\cap C_T({\overline{Q}}_T)$ and

$${\iint }_{Q_T}(-u\frac{\partial \phi }{\partial t}+a_{ij}(x,t,u)D_{i}u{D}_j\phi -\beta (u)\cdot \mathrm{\nabla }\phi -B(x,t,u)\phi -h(x,t)\phi )dxdt=0$$

(2.1)

for any $\phi \in C^1({\overline{Q}}_T)$ with $\phi (x,0)=\phi (x,T)$ and $\phi {|}_{\partial \mathrm{\Omega }\times (0,T)}=0$, where $\beta (u)=({\beta }_1(u),\dots ,{\beta }_N(u))$ and ${\beta }_i(u)={\int }_0^u{b}_i(s)ds$, $i=1,\dots ,N$.

For convenience, we let ${\parallel \cdot \parallel }_p$ and ${\parallel \cdot \parallel }_{m,p}$ denote $L^p(\mathrm{\Omega })$ and $W^{m,p}(\mathrm{\Omega })$ norms, respectively. First, we establish the following a priori estimate which plays an important role in the proof of the main results of this paper.

Lemma 1 Let u be a solution of

(2.2)

(2.3)

(2.4)

with $\sigma \in [0,1]$, then there exists a positive constant R independent of σ such that

$${\parallel u(t)\parallel }_{L^{\mathrm{\infty }}(Q_T)}<R,$$

(2.5)

when the measure of Ω is small enough.

Proof Suppose u is a solution of the problem (2.2)-(2.4). Multiplying equation (2.2) by ${|u|}^{p}u$ ($p\ge 0$) and integrating the resulting relation over Ω, noticing that

$$\begin{array}{rcl}{\int }_{\mathrm{\Omega }}b(u)\cdot \mathrm{\nabla }u{|u|}^{p}udx& =& {\int }_{\mathrm{\Omega }}\sum _{i=1}^N{b}_i(u){|u|}^{p}u\frac{\partial u}{\partial x_i}dx=\sum _{i=1}^N{\int }_{\mathrm{\Omega }}{({\int }_0^u{b}_i(s){|s|}^{p}sds)}_{x_i}dx\\ =& \sum _{i=1}^N{\int }_{\partial \mathrm{\Omega }}({\int }_0^u{b}_i(s){|s|}^{p}sds)cos(\mathrm{n},x_i)dx=0,\end{array}$$

where n is the outer normal to ∂ Ω, we have

(2.6)

The second term of the left-hand side in the above integral equality can be written as

$$\begin{array}{rcl}-{\int }_{\mathrm{\Omega }}{D}_i(a_{ij}(x,t,u)D_{j}u){|u(t)|}^{p}u(t)dx& =& (p+1){\int }_{\mathrm{\Omega }}{|u(t)|}^p{a}_{ij}(x,t,u)D_{j}u{D}_{i}udx\\ \ge & \frac{4\lambda (p+1)}{{(p+2)}^2}{\int }_{\mathrm{\Omega }}|\mathrm{\nabla }[{|u(t)|}^{\frac{p}{2}}u(t)]{|}^{2}dx,\end{array}$$

and

$${\int }_{\mathrm{\Omega }}{|u(t)|}^{p}u(t)hdx\le {\left({\int }_{\mathrm{\Omega }}{|u(t)|}^{p+2}dx\right)}^{\frac{p+1}{p+2}}{\left({\int }_{\mathrm{\Omega }}{h}^{p+2}dx\right)}^{\frac{1}{p+2}}.$$

Hence, from (2.6), we have

$$\frac{d}{dt}{\parallel u(t)\parallel }_{p+2}^{p+2}+C_1{\parallel \mathrm{\nabla }({|u(t)|}^{\frac{p}{2}}u(t))\parallel }_2^2\le C_2(p+2)({\parallel u(t)\parallel }_{p+\alpha +1}^{p+\alpha +1}+{\parallel u\parallel }_{p+2}^{p+1}),$$

(2.7)

where $C_1$, $C_2$ are positive constants independent of $u(t)$, p.

If $0\le \alpha <1$, by Hölder’s inequality and Young’s inequality, we have

$$\begin{array}{rcl}{\int }_{\mathrm{\Omega }}{|u(t)|}^{p+\alpha +1}dx& \le & {\left({\int }_{\mathrm{\Omega }}{|u(t)|}^{p+2}dx\right)}^{\frac{p+\alpha +1}{p+2}}{|\mathrm{\Omega }|}^{\frac{1-\alpha }{p+2}}\\ \le & max\{1,{|\mathrm{\Omega }|}^{\frac{1}{2}}\}{\parallel u(t)\parallel }_{p+2}^{p+\alpha +1}\\ =& max\{1,{|\mathrm{\Omega }|}^{\frac{1}{2}}\}{\parallel u(t)\parallel }_{p+2}^{(p+2)\alpha }{\parallel u(t)\parallel }_{p+2}^{(p+1)(1-\alpha )}\\ \le & {\parallel u(t)\parallel }_{p+2}^{p+2}+C{\parallel u(t)\parallel }_{p+2}^{p+1}.\end{array}$$

(2.8)

Combined with (2.7), it yields

$$\frac{d}{dt}{\parallel u(t)\parallel }_{p+2}^{p+2}+C_1{\parallel \mathrm{\nabla }({|u(t)|}^{\frac{p}{2}}u(t))\parallel }_2^2\le C_2(p+2)({\parallel u(t)\parallel }_{p+2}^{p+2}+{\parallel u(t)\parallel }_{p+2}^{p+1}).$$

(2.9)

If $\alpha =1$, from (2.7) we can get (2.9) directly.

Set

$$u_k(t)={|u(t)|}^{\frac{p_k}{2}}u(t),p_k=2^k-2(k=1,2,\dots ),$$

then $p_k=2p_{k-1}+2$. From (2.9), we have

$$\frac{d}{dt}{\parallel u_k(t)\parallel }_2^2+C_1{\parallel \mathrm{\nabla }{u}_k(t)\parallel }_2^2\le C_2(p_k+2){\parallel u_k(t)\parallel }_2^2+C_2(p_k+2){\parallel u_k(t)\parallel }_2^{\frac{2(p_k+1)}{p_k+2}}.$$

By the Gagliardo-Nirenberg inequality, we have

$${\parallel u_k(t)\parallel }_2\le C{\parallel \mathrm{\nabla }{u}_k(t)\parallel }_2^{\theta }{\parallel u_k(t)\parallel }_1^{1-\theta },\text{with }\theta =\frac{N}{N+2}\in (0,1).$$

(2.10)

Noticing ${\parallel u_k(t)\parallel }_1={\parallel u_{k-1}(t)\parallel }_2^2$, by (2.10) we obtain

$$\begin{array}{rcl}\frac{d}{dt}{\parallel u_k(t)\parallel }_2^2& \le & -C_1{\parallel u_k(t)\parallel }_2^{\frac{2}{\theta }}{\parallel u_k(t)\parallel }_1^{\frac{2(\theta -1)}{\theta }}+C_2(p_k+2){\parallel u_k(t)\parallel }_2^2\\ +C_2(p_k+2){\parallel u_k(t)\parallel }_2^{\frac{2(p_k+1)}{p_k+2}}\\ \le & -C_1{\parallel u_k(t)\parallel }_2^{\frac{2}{\theta }}{\parallel u_{k-1}(t)\parallel }_2^{\frac{4(\theta -1)}{\theta }}+C_2(p_k+2){\parallel u_k(t)\parallel }_2^2\\ +C_2(p_k+2){\parallel u_k(t)\parallel }_2^{\frac{2(p_k+1)}{p_k+2}}.\end{array}$$

(2.11)

Set ${\lambda }_k=max\{1,{sup}_t{\parallel u_k(t)\parallel }_2\}$, then

$$\begin{array}{rcl}\frac{d}{dt}{\parallel u_k(t)\parallel }_2^2& \le & {\parallel u_k(t)\parallel }_2^{\frac{2(p_k+1)}{p_k+2}}\{-C_1{\parallel u_k(t)\parallel }_2^{\frac{2}{\theta }-\frac{2(p_k+1)}{p_k+2}}{\lambda }_{k-1}^{\frac{4(\theta -1)}{\theta }}\\ +C_2(p_k+2){\parallel u_k(t)\parallel }_2^{\frac{2}{p_k+2}}+C_2(p_k+2)\}.\end{array}$$

(2.12)

Now, we estimate $(p_k+2){\parallel u_k(t)\parallel }_2^{\frac{2}{p_k+2}}$. By Young’s inequality,

$$ab\le ϵa^{p^{\prime }}+{ϵ}^{-\frac{q^{\prime }}{p^{\prime }}}\frac{1}{q^{\prime }}{\left(\frac{1}{p^{\prime }}\right)}^{\frac{q^{\prime }}{p^{\prime }}}{b}^{q^{\prime }},$$

where $p^{\prime }>1$, $q^{\prime }>1$, $\frac{1}{p^{\prime }}+\frac{1}{q^{\prime }}=1$ with

we have

$$(p_k+2){\parallel u_k(t)\parallel }_2^{\frac{2}{p_k+2}}\le \frac{1}{2}{\parallel u\parallel }_2^{\frac{2}{\theta }-\frac{2(p_k+1)}{p_k+2}}{\lambda }_{k-1}^{\frac{4(\theta -1)}{\theta }}+C{(p_k+2)}^{\frac{l_k}{l_k-1}}{\lambda }_{k-1}^{\frac{4(1-\theta )}{\theta (l_k-1)}}.$$

(2.13)

It is easy to see that ${lim}_{k\to \mathrm{\infty }}{l}_k=+\mathrm{\infty }$. Denote

$$a_k=\frac{l_k}{l_k-1},b_k=\frac{4(1-\theta )}{\theta (l_k-1)}.$$

From (2.12), (2.13), we have

That is,

(2.14)

The periodicity of $u_k(t)$ implies that there exists $t^{\prime }$ such that ${\parallel u_k(t)\parallel }_2$ takes its maximum and the left-hand side of (2.14) vanishes. Then we have

$${\parallel u_k(t)\parallel }_2\le {\left\{C[(p_k+2)+{(p_k+2)}^{a_k}{l}_{k-1}^{b_k}]{\lambda }_{k-1}^{\frac{4(1-\theta )}{\theta }}\right\}}^{\frac{1}{c_k}},$$

where

$$c_k=\frac{2}{\theta }-\frac{2(p_k+1)}{p_k+2}=\frac{2l_k}{p_k+2}.$$

Since ${\lambda }_{k-1}\ge 1$ ($k=1,2,\cdot$), it follows that

$${\parallel u_k(t)\parallel }_2\le {\left\{C{(p_k+2)}^{a_k}{\lambda }_{k-1}^{b_k+\frac{4(1-\theta )}{\theta }}\right\}}^{\frac{1}{c_k}}={\left\{C{(p_k+2)}^{a_k}\right\}}^{\frac{p_k+2}{2l_k}}{\lambda }_{k-1}^{\frac{4(1-\theta )(p_k+2)}{2(l_k-1)\theta }}.$$

Noticing that $\frac{p_k+2}{(l_k-1)\theta }=\frac{1}{1-\theta }$ and $\frac{p_k+2}{2l_k}$ are bounded, we have

$${\parallel u_k(t)\parallel }_2\le C{2}^{k{a}^{\prime }}{\lambda }_{k-1}^2,$$

where $a^{\prime }$ is a positive constant independent of k. That is,

$$ln{\parallel u_k(t)\parallel }_2\le ln{l}_k\le lnC+klnA+2ln{l}_{k-1},$$

where $A=2^{a^{\prime }}>1$, then

$$\begin{array}{rcl}ln{\parallel u_k(t)\parallel }_2& \le & lnC\sum _{i=0}^{k-2}{2}^i+2^{k-1}ln{\lambda }_1+lnA(\sum _{j=0}^{k-2}(k-j)2^j)\\ \le & (2^{k-1}-1)lnC+2^{k-1}ln{\lambda }_1+f(k)lnA\end{array}$$

with

$$f(k)=2^{k+1}-2^{k-1}-k-2.$$

That is,

$${\parallel u(t)\parallel }_{p_k+2}\le {\left\{C^{2^{k-1}}{\lambda }_1^{2^{k-1}}{A}^{f(k)}\right\}}^{\frac{2}{p_k+2}}.$$

Letting $k\to \mathrm{\infty }$, we obtain

$${\parallel u(t)\parallel }_{\mathrm{\infty }}\le C{\lambda }_1^2\le C{(max\{1,\underset{t}{sup}{\parallel u(t)\parallel }_2\})}^2.$$

(2.15)

In order to estimate ${\parallel u(t)\parallel }_2$, we set $p=0$. From (2.9), we obtain

$$\frac{d}{dt}{\parallel u(t)\parallel }_2^2+C_1{\parallel \mathrm{\nabla }u(t)\parallel }_2^2\le C_2{\parallel u(t)\parallel }_2^2+C_2{\parallel u(t)\parallel }_2.$$

According to the Poincaré inequality, we have

$$C_p{\parallel u(t)\parallel }_2^2\le {\parallel \mathrm{\nabla }u(t)\parallel }_2^2,$$

where $C_p$ is a positive constant which depends only on N and the measure of Ω and becomes very large when $|\mathrm{\Omega }|$ becomes small. Then

$$\frac{d}{dt}{\parallel u(t)\parallel }_2^2+C_1{C}_p{\parallel u(t)\parallel }_2^2\le C_2{\parallel u(t)\parallel }_2^2+C_2{\parallel u(t)\parallel }_2.$$

So, when $|\mathrm{\Omega }|$ is sufficiently small, we have $C_1{C}_p>C_2$. Then by Young’s inequality, we obtain

$$\frac{d}{dt}{\parallel u(t)\parallel }_2^2+C{\parallel u(t)\parallel }_2^2\le C,$$

where C is a constant independent of u. By the periodicity of u, we have

$${\parallel u(t)\parallel }_2\le R,$$

where R is a positive constant independent of σ. Combining the above inequality with (2.15), we obtain (2.5). The proof is completed. □

3 The main results

Our main result is the following theorem.

Theorem 1 If (A1), (A2) and (A3) hold, then the problem (1.1)-(1.3) admits at least one periodic solution u.

Proof First, we define a map by considering the following problem:

(3.1)

(3.2)

(3.3)

where $f(x,t)$ is a given function in $C_T({\overline{Q}}_T)$. It follows from a standard argument similar to [10] that the problem (3.1)-(3.3) admits a unique solution. So, we can define a map $T:C_T({\overline{Q}}_T)\to C_T({\overline{Q}}_T)$ by $u=Tf$ and the map $u=Tf$ is compact and continuous. In fact, by the method in [9], we can infer that ${\parallel u\parallel }_{L^{\mathrm{\infty }}(Q_T)}$ is bounded if $f\in L^{\mathrm{\infty }}(Q_T)$ and u, $\mathrm{\nabla }u\in C^{\alpha }(\overline{Q_T})$ for some $\alpha >0$. Then (by the Arzela-Ascoli theorem) the compactness of the map T comes from ${\parallel u\parallel }_{L^{\mathrm{\infty }}(Q_T)}$ and the Hölder continuity of u. The continuity of the map T comes from the Hölder continuity of ∇u.

Let $\mathrm{\Phi }(u)=B(x,t,u)+h(x,t)$, by (A2)-(A3) and the above arguments, we see that $T(\sigma \mathrm{\Phi })$ is a complete continuous map for $\sigma \in [0,1]$. By Lemma 1, we can see that any fixed point u of the map $T(\sigma \mathrm{\Phi })$ satisfies

$${\parallel u\parallel }_{\mathrm{\infty }}\le C,$$

where C is a positive constant independent of σ. Then, by the Leray-Schauder fixed point theorem [13], we conclude that the problem (1.1)-(1.3) admits a periodic solution u. The proof is complete. □