Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent
© Li and Gao; licensee Springer. 2014
Received: 19 October 2013
Accepted: 20 March 2014
Published: 2 April 2014
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.
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.
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.
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)
- 1.space: We haveequipped with the following Luxemburg norm:
The space is a separable, uniformly convex Banach space.
- 2.space: We have
endowed with the norm . We denote by the closure of in . In fact, the norm and are equivalent norms in . and are separable and reflexive Banach spaces with the above norms.
- 3.Frequently used relationships in the estimate:
-Hölder’s inequality:For any and , with , we have
Embedding relationships:If and are in , and , for any , then there exists a positive constant such that
i.e. the embedding is continuous.If and , for any , then the embedding is continuous and compact. Here
There exists a positive constant such that , for any .
We next define the weak solutions to problem (1.1).
for all satisfying for and for .
where , and are given constants.
for all satisfying for and for .
Remark 2.3 For any , the set is dense in , thus in the sense of the definition of weak solution above, can be chosen as test function.
Therefore, if a nonnegative function satisfies , then is a weak solution of (2.2).
Lemma 2.4 Assume that , and . Then is a continuous compact operator from to . Furthermore, .
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)
where , and .
Next, we prove a crucial a priori bound for via a De Giorgi iteration technique as in .
Then there exists a constant , such that , where R is independent of ϵ and η.
for any , where depends on q, , m, and Ω.
for any , where C depends on q, , m, T and Ω.
From (3.26) and (3.27), we get . □
Proposition 3.4 Assume that , . If solves , for some and , then for any . Moreover, if , then in .
In what follows, we prove a lower bound for the regularized problem.
is the embedding constant of into , and is the Lebesgue measure of the domain Ω.
Proof We argue by contradiction. If not, then for each and , there exists a such that , with . For clarity, we divide the proof into four steps.
which is clearly a contradiction to the assumption that . This completes the proof. □
Theorem 3.6 Let be as given in Proposition 3.5. Then for all .
Also, from Proposition 3.5, we infer that admits no nontrivial solution in . Moreover, is not a solution of . So . Together with (3.39), we have . □
4 Existence of nontrivial nonnegative solution to (1.1)
Theorem 4.1 Assume for a.e. and . Then problem (1.1) has a nontrivial nonnegative periodic solution.
for and . Using the solvability of the Leray-Schauder degree, we conclude that the regularized problem (2.2) admits a nontrivial nonnegative solution in .
We prove that with and that a solution to problem (1.1) is obtained as a limit of as . We proceed in several steps.
So and is uniformly bounded in the space . Thus, up to subsequence if necessary, we may assume that . In what follows, our main goal is to prove that u is a weak solution of problem (1.1).
weakly in as .
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 .
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.
We choose and . Consequently, these choices guarantee . From (5.2) and is nonnegative and nonincreasing, we have deduced the result. □
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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