Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent

Here, Ω is a bounded simply connected domain with smooth boundary ∂ Ω in ${\mathbb{R}}^N$, $Q_T=\mathrm{\Omega }\times (0,T)$, ${\mathrm{\Gamma }}_T=\partial \mathrm{\Omega }\times (0,T)$, $T>0$, and $\tau \in (0,+\mathrm{\infty })$. We assume $m>1$, $p\in C^{1,\alpha }(\overline{\mathrm{\Omega }})$, with $p^{+}:={max}_{\overline{\mathrm{\Omega }}}p(x)$, $p^{-}:={min}_{\overline{\mathrm{\Omega }}}p(x)$, $p^{+}\ge p^{-}>2$ and that $a\in L^{\mathrm{\infty }}(Q_T)$ and $K\in L^{\mathrm{\infty }}(Q_T)$ can be extended as T-periodic functions to $\mathrm{\Omega }\times R$. Furthermore, we assume that $K\ge 0$ for a.e. $(x,t)\in Q_T$.

Equation (1.1) is a doubly degenerate parabolic equation with delay and nonlocal term, which models diffusive periodic phenomena in physics and mathematical biology. In biology, it arises from population model, where $u(x,t)$ denotes the density of population at time t located at $x\in \mathrm{\Omega }$, a is the natural growth rate of the population, the nonlocal term ${\int }_{\mathrm{\Omega }}K(\xi ,t)u^2(\xi ,t-\tau )d\xi$ evaluates a weighted fraction of individual, and the delayed density u at time $t-\tau$ appearing in the nonlocal term represents the time needed to an individual to become adult. In physics, problem (1.1) is proposed based on some evolution phenomena in electrorheological fluids [1]. It describes the ability of a conductor to undergo significant changes when an electric field is imposed on. This model has been employed for some technological applications, such as medical rehabilitation equipment and shock wave absorber.

They proved the existence of a nontrivial nonnegative periodic solution via monotonicity method. Using a Moser iterative method (see [8–11]), they also obtained some a priori bounds and asymptotic behaviors for the solutions.

Recently, the variable exponent Sobolev space and its applications have attracted considerable interest; see [1, 12–14] and the references therein. When $p^{+}\ge p^{-}>2$ and $m>1$, the doubly degenerate parabolic equation (1.1) is a more realistic model which describes the rather slow diffusion process. In our models, the principal term $div(|\mathrm{\nabla }{u}^m{|}^{p(x)-2}\mathrm{\nabla }{u}^m)$, in place of the usual term $\mathrm{△}{u}^m$ or $div(|\mathrm{\nabla }{u}^m{|}^{p-2}\mathrm{\nabla }{u}^m)$, represents nonhomogeneous diffusion that depends on the position $x\in \mathrm{\Omega }$ and thus gives a better description of nonhomogeneous character of the process.

There are many differences between Sobolev spaces with constant exponent and those with variable exponent; many powerful tools applicable in constant exponent spaces are not available for variable exponent spaces. For instance, the variable exponent spaces are no longer translation invariant and Young’s inequality ${\parallel f\ast g\parallel }_{p(\cdot )}\le C{\parallel f\parallel }_{p(\cdot )}{\parallel g\parallel }_1$ holds if and only if p is constant (see monograph [12]). As we all know, the frequently used Hölder’s inequality, Poincaré’s inequality, etc., will be presented in new forms for variable exponent spaces.

The presence of the nonlocal term and $p(x)$-Laplacian term makes the sup-solution and sub-solution method (as in [5]) in vain. In our paper, we adopt the topological degree method (as in [8–10]) to show the existence of nontrivial periodic solutions to problem (1.1). However, the method employed in the variable exponent case [13] or in the constant exponent case [8–11] cannot be directly used to derive the uniform upper bound for solutions, which is a crucial step in applying the topological degree method. We apply a modified De Giorgi iteration to establish the crucial uniform bound. We believe that the modified De Giorgi iteration used in this paper can be employed to other types equations with nonstandard growth conditions.

In Section 3, in order to apply the topological degree method, we combine these regularized problems with a relatively simpler equation and derive some a priori estimates. By virtue of the De Giorgi iteration technique, we deduce an a priori $L^{\mathrm{\infty }}$ bound for solutions to the regularized problems in Proposition 3.2; and the uniform lower bound estimate is obtained in Proposition 3.5. In Section 4, we establish the existence of nonnegative nontrivial solution of (1.1) through the limit process as ϵ and η tend to zero. Finally, in the Appendix, we give a proof of the iteration lemma (Lemma 3.1) for the sake of readability.

First of all, for the reader’s convenience, we recall some preliminary results concerning the variable exponent Sobolev spaces. One may find these standard results in monographs [1, 12].

There exists a positive constant $C_p$ such that $|u{|}_{L^{p(x)}(\mathrm{\Omega })}\le C_p|\mathrm{\nabla }u{|}_{L^{p(x)}(\mathrm{\Omega })}$, for any $u\in W_0^{1,p(x)}(\mathrm{\Omega })$.

We next define the weak solutions to problem (1.1).

for all $\phi \in C^1({\overline{Q}}_T)$ satisfying $\phi (x,T)=\phi (x,0)$ for $x\in \mathrm{\Omega }$ and $\phi (\cdot ,t){|}_{\partial \mathrm{\Omega }}=0$ for $t\in [0,T]$.

where $0<ϵ<\frac{1}{2}$, $0<\eta <{(\frac{1}{2})}^{\frac{p^{+}-2}{p^{-}-2}}$ and $\sigma \in [0,1]$ are given constants.

for all $\phi \in C^1({\overline{Q}}_T)$ satisfying $\phi (x,T)=\phi (x,0)$ for $x\in \mathrm{\Omega }$ and $\phi (\cdot ,t){|}_{\partial \mathrm{\Omega }}=0$ for $t\in [0,T]$.

Remark 2.3 For any $p\in C^{1,\alpha }(\overline{\mathrm{\Omega }})$, the set $\{\phi \in C^1({\overline{Q}}_T):\phi (\cdot ,T)=\phi (\cdot ,0)\}$ is dense in $\{\phi \in C({\overline{Q}}_T)\cap L^{p^{-}}(0,T;W_0^{1,p(x)}(\mathrm{\Omega })):|\mathrm{\nabla }{u}^m|\in L^{p(x)}(Q_T),\phi (\cdot ,T)=\phi (\cdot ,0)\}$, thus in the sense of the definition of weak solution above, $u_{ϵ\eta }$ can be chosen as test function.

Therefore, if a nonnegative function $u_{ϵ\eta }\in L^{\mathrm{\infty }}(Q_T)$ satisfies $u_{ϵ\eta }=G_{ϵ\eta }(1,f(u_{ϵ\eta }))$, then $u_{ϵ\eta }$ is a weak solution of (2.2).

Then, according to [3] or the classical regularity results from [4], one obtains the following lemma.

Lemma 2.4 Assume that $0<ϵ<\frac{1}{2}$, $0<\eta <{(\frac{1}{2})}^{\frac{p^{+}-2}{p^{-}-2}}$ and $\lambda \in L^{\mathrm{\infty }}(Q_T)$. Then $T_{ϵ\eta }$ is a continuous compact operator from $[0,1]\times L^{\mathrm{\infty }}(Q_T)$ to $L^{\mathrm{\infty }}(Q_T)$. Furthermore, $u_{ϵ\eta }=T_{ϵ\eta }(\sigma ,\lambda )\in C({\overline{Q}}_T)$.

First of all, the following modified De Giorgi iteration lemma will be useful (we give a proof in the Appendix).

Lemma 3.1 (Iteration lemma)

where $d=M{2}^{\frac{\delta }{\delta -1}}{({\phi }^{\beta -1}(k_0)+{\phi }^{\gamma -1}(k_0))}^{\frac{1}{\alpha }}$, and $\delta =min\{\beta ,\gamma \}$.

Next, we prove a crucial a priori $L^{\mathrm{\infty }}$ bound for $u_{ϵ\eta }$ via a De Giorgi iteration technique as in [15].

Then there exists a constant $R>0$, such that ${\parallel u_{ϵ\eta }\parallel }_{L^{\mathrm{\infty }}(Q_T)}<R$, where R is independent of ϵ and η.

for any $q>1$, where $C_1$ depends on q, $p^{-}$, m, and Ω.

for any $q>1$, where C depends on q, $p^{-}$, m, T and Ω.

 □

where $u^{+}=max\{u,0\}$.

From (3.26) and (3.27), we get $deg(u-T_{ϵ\eta }(1,u^{+}),B_R,0)=1$. □

Using the standard method, similar to that in [3] or [13], one can prove the following.

Proposition 3.4 Assume that $a\in L^{\mathrm{\infty }}(Q_T)$, $K\in L^{\mathrm{\infty }}(Q_T)$. If $u_{ϵ\eta }$ solves $u=G_{ϵ\eta }(\sigma ,\rho f(u^{+})+1-\sigma )$, for some $\sigma \in [0,1]$ and $\rho \in [0,1]$, then $u_{ϵ\eta }\ge 0$ for any $(x,t)\in Q_T$. Moreover, if $u_{ϵ\eta }\ne 0$, then $u_{ϵ\eta }>0$ in $Q_T$.

In what follows, we prove a lower bound for the regularized problem.