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
In this paper, we are interested in the following evolutional -Laplacian equation:
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)
Let p be a continuous function defined in , , for any .
1. space: We have
equipped 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:
4. -Hölder’s inequality:
For any and , with , we have
5. 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
6. -Poincaré’s inequality:
There exists a positive constant such that , for any .
We next define the weak solutions to problem (1.1).
Definition 2.1u is said to be a weak periodic solution to (1.1) provided that with , and u satisfies
for all satisfying for and for .
As in , we introduce the following regularized problem:
where , and are given constants.
Definition 2.2 We say that is a weak periodic solution of (2.2), if with , , and solves
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.
We investigate problem (2.2) extensively before studying the limit process as . Define a map as follows:
where is a weak periodic solution of the problem:
Given , let be defined by
Therefore, if a nonnegative function satisfies , then is a weak solution of (2.2).
Lemma 2.4Assume 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)
Suppose is a nonnegative and nonincreasing function on , it satisfies
for any , and for some constants , , , . Then
where , and .
Next, we prove a crucial a priori bound for via a De Giorgi iteration technique as in .
Proposition 3.2Let and assume that is a nonnegativeT-periodic continuous function such that
Then there exists a constant , such that , whereRis independent ofϵandη.
ProofStep 1. Multiplying (3.2) by , with any . Integrating over Ω and noticing that , we have
Since , 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 and appropriately, we have from (3.6) and (3.7)
for any , where depends on q, , m, and Ω.
Integrating (3.6) over and using the T-periodicity of , we have
Similarly to (3.7), we obtain
where depends on q, , m, T and Ω. By Poincaré’s inequality, we have
Recall our assumption that , , and thus . Consequently, considering (3.11), we obtain
which implies that there exists a such that
From (3.8) and (3.13), we conclude
for any . In view of the T-periodicity of , (3.14) shows
We finally arrive at
for any , where C depends on q, , m, T and Ω.
Step 2. Let
where is the Lebesgue measure of the set . Multiplying (3.2) by on both sides, where represents the characteristic function of the interval , and integrating over , we have
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
Let . 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 , it is easy to see
The relationships (3.23) and (3.24) above imply that
Noticing that and , by the iteration Lemma 3.1, we obtain and thus , where
Theorem 3.3Assume , for a.e. . Then there exists a positive constantRsuch 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
Using the fact that , one has
From (3.26) and (3.27), we get . □
Proposition 3.4Assume 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.
Proposition 3.5Let be the first eigenvalue of
and let be the associated positive eigenfunction such that . Assume that , and . If satisfies for some , then , where
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.
Step 1. Note that, by Proposition 3.4, in . Taking and multiplying
by , integrating over and using the T-periodicity of , we obtain
Step 2. Using , we have
Since and , we have . Hence
Thanks to the -Hölder’s inequality in variable exponent space, we have
Noting that and using Hölder’s inequality, we have
Integrating (3.31) over and noting (3.32), we get
Step 3. Multiplying (3.28) by , integrating over , noticing the T-periodicity of and , we deduce
Substituting this inequality into (3.33), we have
Substituting (3.34) into (3.30) and noticing that , we get
Considering that , from (3.29) and (3.35), we have
Step 4. We claim
from which we will derive a contradiction. First, to show (3.37), let in (3.36). Using the fact that and noting and , we get
Now the definition of and (3.37) yield
which is clearly a contradiction to the assumption that . This completes the proof. □
Theorem 3.6Let be as given in Proposition 3.5. Then for all .
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
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 4.1Assume for a.e. and . Then problem (1.1) has a nontrivial nonnegative periodic solution.
Proof We consider the regularized problem (2.2). By Theorem 3.3 and Theorem 3.6, we conclude that there exist R and r, independent of ϵ and η, with , such that
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.
Step 1. In view of , choosing , we have
Multiplying (4.1) by , integrating over and noting the T-periodicity of and the boundedness of , we have
where M is a positive constant independent of ϵ and η. Moreover,
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).
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
Integrating (4.6) over and using Hölder’s inequality, we have
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