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

The authors investigate a degenerate parabolic equation with delay and nonlocal term, which describes slow diffusive processes in physics or biology. The existence of a nonnegative nontrivial periodic solution is obtained through the use of the Leray-Schauder degree method.

MSC:35D05, 35K55.

In this paper, we are interested in the following evolutional p(x)-Laplacian equation:

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.

When p(x) is a constant and m>1, p>2, the model describes the slow diffusion process in physics, which has been extensively investigated; see [2–7]. For example, in [5], the authors studied the following doubly degenerate parabolic equation with logistic periodic sources:

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.

We now discuss the main plan of the paper. In Section 2, we review some preliminaries concerning the variable exponent Sobolev spaces and introduce a family of regularized problems for problem (1.1). We regularize the degenerate part through replacing the term div(|\mathrm{\nabla }{u}^m{|}^{p(x)-2}\mathrm{\nabla }{u}^m) by

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].

Let p be a continuous function defined in \overline{\mathrm{\Omega }}, p(x)>1, for any x\in \overline{\mathrm{\Omega }}.

1. 1.

   L^{p(x)}(\mathrm{\Omega }) space: We have

   equipped with the following Luxemburg norm:

   |u{|}_{L^{p(x)}(\mathrm{\Omega })}:=inf\{\lambda >0:{\int }_{\mathrm{\Omega }}|\frac{u(x)}{\lambda }{|}^{p(x)}dx\le 1\}.

   The space (L^{p(x)}(\mathrm{\Omega }),|\cdot {|}_{L^{p(x)}(\mathrm{\Omega })}) is a separable, uniformly convex Banach space.

2. 2.

   W^{1,p(x)}(\mathrm{\Omega }) space: We have

   W^{1,p(x)}(\mathrm{\Omega }):=\{u\in L^{p(x)}(\mathrm{\Omega }):|\mathrm{\nabla }u|\in L^{p(x)}(\mathrm{\Omega })\},

   endowed with the norm |u{|}_{W^{1,p(x)}(\mathrm{\Omega })}:=|\mathrm{\nabla }u{|}_{L^{p(x)}(\mathrm{\Omega })}+|u{|}_{L^{p(x)}(\mathrm{\Omega })}. We denote by W_0^{1,p(x)}(\mathrm{\Omega }) the closure of C_0^{\mathrm{\infty }}(\mathrm{\Omega }) in W^{1,p(x)}(\mathrm{\Omega }). In fact, the norm |\mathrm{\nabla }u{|}_{L^{p(x)}(\mathrm{\Omega })} and {|u|}_{W^{1,p(x)}(\mathrm{\Omega })} are equivalent norms in W_0^{1,p(x)}(\mathrm{\Omega }). W^{1,p(x)}(\mathrm{\Omega }) and W_0^{1,p(x)}(\mathrm{\Omega }) are separable and reflexive Banach spaces with the above norms.

3. 3.

   Frequently used relationships in the estimate:

   \begin{array}{r}min\{{\left({\int }_{\mathrm{\Omega }}\right|u(x){|}^{p(x)}dx)}^{\frac{1}{p^{+}}},{\left({\int }_{\mathrm{\Omega }}\right|u(x){|}^{p(x)}dx)}^{\frac{1}{p^{-}}}\}\\ \le |u{|}_{L^{p(x)}(\mathrm{\Omega })}\\ \le max\{{\left({\int }_{\mathrm{\Omega }}\right|u(x){|}^{p(x)}dx)}^{\frac{1}{p^{+}}},{\left({\int }_{\mathrm{\Omega }}\right|u(x){|}^{p(x)}dx)}^{\frac{1}{p^{-}}}\}.\end{array}
4. 4.

   p(x) -Hölder’s inequality:

   For any u\in L^{p(x)}(\mathrm{\Omega }) and v\in L^{q(x)}(\mathrm{\Omega }), with \frac{1}{p(x)}+\frac{1}{q(x)}=1, we have

   \left|{\int }_{\mathrm{\Omega }}uvdx\right|\le (\frac{1}{p^{-}}+\frac{1}{q^{-}})|u{|}_{L^{p(x)}(\mathrm{\Omega })}|v{|}_{L^{q(x)}(\mathrm{\Omega })}.
5. 5.

   Embedding relationships:

   If p_1 and p_2 are in C(\overline{\mathrm{\Omega }}), and 1\le p_1(x)\le p_2(x), for any x\in \overline{\mathrm{\Omega }}, then there exists a positive constant C_{p_1(x),p_2(x)} such that

   |u{|}_{L^{p_1(x)}(\mathrm{\Omega })}\le C_{p_1(x),p_2(x)}|u{|}_{L^{p_2(x)}(\mathrm{\Omega })},

   i.e. the embedding L^{p_2(x)}(\mathrm{\Omega })\hookrightarrow L^{p_1(x)}(\mathrm{\Omega }) is continuous.

   If q\in C(\overline{\mathrm{\Omega }}) and , for any x\in \overline{\mathrm{\Omega }}, then the embedding W_0^{1,p(x)}(\mathrm{\Omega })\hookrightarrow L^{q(x)}(\mathrm{\Omega }) is continuous and compact. Here
6. 6.

   p(x) -Poincaré’s inequality:

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).

Definition 2.1 u is said to be a weak periodic solution to (1.1) provided that u^m\in L^{p^{-}}(0,T;W_0^{1,p(x)}(\mathrm{\Omega })) with |\mathrm{\nabla }{u}^m|\in L^{p(x)}(Q_T), u\in C({\overline{Q}}_T) and u satisfies

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].

As in [7], we introduce the following regularized problem:

where , and \sigma \in [0,1] are given constants.

Definition 2.2 We say that u_{ϵ\eta } is a weak periodic solution of (2.2), if u_{ϵ\eta }^m\in L^{p^{-}}(0,T;W_0^{1,p(x)}(\mathrm{\Omega })) with |\mathrm{\nabla }{u}_{ϵ\eta }^m|\in L^{p(x)}(Q_T), u_{ϵ\eta }\in C({\overline{Q}}_T), and u_{ϵ\eta } solves

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.

We investigate problem (2.2) extensively before studying the limit process as ϵ,\eta \to 0. Define a map G_{ϵ\eta }:[0,1]\times L^{\mathrm{\infty }}(Q_T)\to L^{\mathrm{\infty }}(Q_T) as follows:

where u_{ϵ\eta } is a weak periodic solution of the problem:

Given \alpha \in L^{\mathrm{\infty }}(Q_T), let f=f(\alpha )\in L^{\mathrm{\infty }}(Q_T) be defined by

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).

Define

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

Lemma 2.4 Assume that , 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)

Suppose \phi (t) is a nonnegative and nonincreasing function on [K_0,+\mathrm{\infty }), it satisfies

for any h>k\ge k_0, and for some constants M>0, \alpha >0, \beta >1, \gamma >1. Then

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].

Proposition 3.2 Let K_1>0 and assume that u_{ϵ\eta } is a nonnegative T-periodic continuous function such that

Then there exists a constant R>0, such that , where R is independent of ϵ and η.

Proof Step 1. Multiplying (3.2) by u_{ϵ\eta }^{mq}, with any q>1. Integrating over Ω and noticing that u_{ϵ\eta }(\cdot ,t){|}_{\partial \mathrm{\Omega }}=0, we have

Since |\mathrm{\nabla }{u}_{ϵ\eta }^m{|}^{p(x)}\ge |\mathrm{\nabla }{u}_{ϵ\eta }^m{|}^{p^{-}}-1, we deal with the second term on the left-hand side of (3.4) as follows.

Combining (3.4) and (3.5), we have

We estimate the right-hand side of (3.6) by Hölder’s inequality, the embedding theorem and Young’s inequality with ϵ to deduce

Choosing {ϵ}_1 and {ϵ}_2 appropriately, we have from (3.6) and (3.7)

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

Integrating (3.6) over [\tau ,\tau +T] and using the T-periodicity of u_{ϵ\eta }, we have

Similarly to (3.7), we obtain

where C_2 depends on q, p^{-}, m, T and Ω. By Poincaré’s inequality, we have

Recall our assumption that p^{-}>2, m>1, and thus m(p^{-}+q-1)>mq+1. Consequently, considering (3.11), we obtain

which implies that there exists a t_0\in (\tau ,\tau +T) such that

From (3.8) and (3.13), we conclude

for any t\ge t_0. In view of the T-periodicity of u_{ϵ\eta }, (3.14) shows

We finally arrive at

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

Step 2. Let

where |A_k(t)| is the Lebesgue measure of the set A_k(t). Multiplying (3.2) by {(u_{ϵ\eta }-k)}_{+}^m{\chi }_{[t_1,t_2]}(t) on both sides, where {\chi }_{[t_1,t_2]}(t) represents the characteristic function of the interval [t_1,t_2], and integrating over Q_T, we have

Let I_k(t):={\int }_{\mathrm{\Omega }}{(u_{ϵ\eta }-k)}_{+}^{m+1}dx. We assume that the absolutely continuous function I_k(t) attains its maximum at \varrho \in [\tau ,\tau +T]. Take t_1=\varrho -\theta, t_2=\varrho and θ small enough so that t_1>\tau. (In fact, this is always possible because of the periodicity of u_{ϵ\eta }; for example, if \varrho =\tau, we take \varrho =\tau +T, then \varrho >\tau and t_1>\tau.) Then we have I_k(\varrho )\ge I_k(\varrho -\theta ) and

Letting \theta \to 0^{+} yields

After a direct computation, we obtain an estimate for the left-hand side of (3.17) as follows:

Substituting (3.18) into (3.17), we have

We now deal with (3.19). On one hand, by the embedding theorem

where S is the Sobolev embedding constant, and

On the other hand, from (3.15), where we may fix a special q, using Hölder’s inequality, we obtain

Let J_k(\varrho )={\int }_{A_k(\varrho )}{(u_{ϵ\eta }-k)}^{mr}dx. Then (3.19), (3.20), and (3.21) imply

Utilizing Young’s inequality with ϵ, we obtain from (3.22)

Upon choosing ϵ appropriately, one obtains

For any h>k>0, it is easy to see

The relationships (3.23) and (3.24) above imply that

Noticing that \frac{Nr-p^{-}(N+p^{-})}{(p^{-}-1)(N+p^{-})}>0 and \frac{r}{p^{-}}>1, by the iteration Lemma 3.1, we obtain {\mu }_R=0 and thus , where

with

 □

Theorem 3.3 Assume K\ge 0, for a.e. (x,t)\in Q_T. Then there exists a positive constant R such that

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

Proof From Proposition 3.2, we take K_1={\parallel a\parallel }_{L^{\mathrm{\infty }}(Q_T)}, it implies that there exists a positive constant R>0 independent of ϵ and η, such that u_{ϵ\eta }\ne G_{ϵ\eta }(1,\rho f(u_{ϵ\eta }^{+})), for any u_{ϵ\eta }\in \partial B_R, \rho \in [0,1]. Hence the topological degree deg(u-G_{ϵ\eta }(1,\rho f(u^{+})),B_R,0) is well defined in B_R. Thanks to the homotopy invariance property of the Leray-Schauder degree, we have

Using the fact that G_{ϵ\eta }(1,0)=0, one has

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.

Proposition 3.5 Let {\mu }_1 be the first eigenvalue of

and let e_1 be the associated positive eigenfunction such that {\parallel e_1\parallel }_{L^2(\mathrm{\Omega })}=1. Assume that \frac{1}{T}{\iint }_{Q_T}a{e}_1^{2}dxdt>{\mu }_1, and . If u_{ϵ\eta }\ne 0 satisfies u_{ϵ\eta }=G_{ϵ\eta }(\sigma ,f(u_{ϵ\eta }^{+})+1-\sigma ) for some \sigma \in [0,1], then {\parallel u_{ϵ\eta }\parallel }_{L^{\mathrm{\infty }}(Q_T)}\ge r_0, where

C_{p^{+},p(x)} is the embedding constant of L^{p^{+}}(\mathrm{\Omega }) into L^{p(x)}(\mathrm{\Omega }), and |\mathrm{\Omega }| is the Lebesgue measure of the domain Ω.

Proof We argue by contradiction. If not, then for each \sigma \in [0,1] and r\in (0,r_0), there exists a u_{ϵ\eta }\ne 0 such that u_{\epsilon \eta }=G_{ϵ\eta }(\sigma ,f(u_{\epsilon \eta }^{+})+1-\sigma ), with {\parallel u_{ϵ\eta }\parallel }_{L^{\mathrm{\infty }}(Q_T)}\le r. For clarity, we divide the proof into four steps.

Step 1. Note that, by Proposition 3.4, u_{ϵ\eta }>0 in Q_T. Taking \varphi \in C_0^{\mathrm{\infty }}(\mathrm{\Omega }) and multiplying

by \frac{{\varphi }^2}{u_{ϵ\eta }}, integrating over Q_T and using the T-periodicity of u_{ϵ\eta }, we obtain