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.
Keywords:degenerate parabolic equation; periodic solution; variable exponent; topological degree; De Giorgi iteration
Here, Ω is a bounded simply connected domain with smooth boundary ∂Ω in , , , , and . We assume , , with , , and that and can be extended as T-periodic functions to . Furthermore, we assume that for a.e. .
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 denotes the density of population at time t located at , a is the natural growth rate of the population, the nonlocal term evaluates a weighted fraction of individual, and the delayed density u at time 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 . 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 is a constant and , , the model describes the slow diffusion process in physics, which has been extensively investigated; see [2-7]. For example, in , 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 and , 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 , in place of the usual term or , represents nonhomogeneous diffusion that depends on the position 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 holds if and only if p is constant (see monograph ). 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 -Laplacian term makes the sup-solution and sub-solution method (as in ) 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  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 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 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.
2 Preliminaries and the regularized problems of (1.1)
equipped with the following Luxemburg norm:
3. Frequently used relationships in the estimate:
5. Embedding relationships:
We next define the weak solutions to problem (1.1).
As in , we introduce the following regularized problem:
3 A priori estimates to the regularized problem
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)
Next, we prove a crucial a priori bound for via a De Giorgi iteration technique as in .
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
Similarly to (3.7), we obtain
From (3.8) and (3.13), we conclude
We finally arrive at
Step 2. Let
Let . We assume that the absolutely continuous function attains its maximum at . Take , and θ small enough so that . (In fact, this is always possible because of the periodicity of ; for example, if , we take , then and .) Then we have and
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
Utilizing Young’s inequality with ϵ, we obtain from (3.22)
Upon choosing ϵ appropriately, one obtains
The relationships (3.23) and (3.24) above imply that
Proof From Proposition 3.2, we take , it implies that there exists a positive constant independent of ϵ and η, such that , for any , . Hence the topological degree is well defined in . Thanks to the homotopy invariance property of the Leray-Schauder degree, we have
In what follows, we prove a lower bound for the regularized problem.
Substituting this inequality into (3.33), we have
Step 4. We claim
Proof In view of Proposition 3.5, for any fixed , we have proved that for all , . So the Leray-Schauder topological degree is well defined for all . Thanks to the homotopy invariance of the topological degree, we have
4 Existence of nontrivial nonnegative solution to (1.1)
where M is a positive constant independent of ϵ and η. Moreover,
Step 2. The following relation is obvious:
From (4.2) and (4.4), we have
Owing to the embedding results in the variable exponent space, one has
From (4.5) and (4.7), there exists a positive constant C independent of ϵ and η, such that
In the following, we prove
A straightforward computation shows that
Step 3. Using a method analogous to , we get , where C is independent of ϵ and η. Since is uniformly bounded in , and , by compactness theorem (Corollary 4 in ), it follows that in . Thus, we have
for any satisfying for and for (and hence, by density, for any with and T-periodicity). The continuity of u follows from similar Hölder estimates in .
Multiplying the equation
Thus, (4.17) and (4.18) imply
Combining (4.19) with (4.20), we obtain
From (4.23) and (4.24) we have (4.15). This completes the proof of Theorem 4.1. □
In this appendix, we prove Lemma 3.1 for the reader’s convenience.
Proof of Lemma 3.1 Define the following sequence:
where d is to be determined later. Then (3.1) implies the recursive relationship
By induction, one has
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and approved the final version.
The authors would like to thank the anonymous referees for their valuable comments on and suggestions regarding the original manuscript. This work was supported by National Science Foundation of China (11271154), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University.
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