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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 578345, 20 pages
http://dx.doi.org/10.1155/2013/578345
Research Article

Existence and Asymptotic Behavior of Traveling Wave Fronts for a Time-Delayed Degenerate Diffusion Equation

Department of Mathematics, South China University of Technology, Guangzhou 510640, China

Received 15 November 2012; Revised 15 February 2013; Accepted 18 February 2013

Academic Editor: Peixuan Weng

Copyright © 2013 Weifang Yan and Rui Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is concerned with traveling wave fronts for a degenerate diffusion equation with time delay. We first establish the necessary and sufficient conditions to the existence of monotone increasing and decreasing traveling wave fronts, respectively. Moreover, special attention is paid to the asymptotic behavior of traveling wave fronts connecting two uniform steady states. Some previous results are extended.

1. Introduction

In this paper, we consider the traveling wave fronts for the following reaction diffusion equation with Hodgkin-Huxley source: where , , , , , is a constant, and for .

In 1952, Hodgkin and Huxley [1] proposed the Hodgkin-Huxley (H-H) equation which describes the propagation of a voltage pulse through the nerve axon of a squid. Recently, more and more attention has been paid to the linear and semilinear parabolic equations with and without time delay; see, for example, [27]. A natural extension of the H-H model is the following linear diffusion equation: For this equation, there have been many interesting results on the existence and stability of the traveling wave solutions, for instance, [810]. By a traveling wave solution, we mean a solution of (3) of the form with the wave speed .

On the other hand, the classical research of traveling waves for the standard linear diffusion equations with various sources has been extended to some degenerate or singular diffusion equations. For example, Aronson [11] considered the following equation: When , the equation degenerates at . Hence, it has a different feature from the case ; that is, if the initial distribution of has compact support, then also has compact support for each . When , , Aronson [11] showed that (4) possesses a unique sharp traveling wave solution with positive wave speed. Hosono [12] solved the existence problem of traveling wave solutions for (4) especially with nonpositive wave speed and discussed the shape of the solutions. Sánchez-Garduño and Maini [13] considered the following degenerate diffusion equation: and obtained the existence of traveling wave solutions of smooth or sharp (oscillatory and monotone) type.

For other papers concerning the traveling wave solutions for degenerate diffusion equations without time delay, see [1422]. From these results, we see that an obvious difference between the linear diffusion equations and the degenerate diffusion equations is that, in the degenerate diffusion case, there may exist traveling wave fronts of sharp type; that is, the support of the solution is bounded above or below, and at the boundary of the support, the derivative of the traveling wave solution is discontinuous. However, in the linear diffusion case, all traveling wave fronts are of smooth type; that is, the solutions are classical solutions, which approach the steady states at infinity.

As far as we know, there are only two articles dealing with the traveling wave solutions for degenerate diffusion equations with time delay. In [23, 24], Jin et al. considered the following time-delayed Newtonian filtration and non-Newtonian filtration equations: By using the shooting method together with the comparison technique, they first obtained the necessary and sufficient conditions to the existence of monotone increasing and decreasing traveling wave solutions, respectively, and then gave an accurate estimation on the convergent rate for the semifinite or infinite traveling waves.

Motivated by [23, 24], in this paper, we discuss the existence and asymptotic behavior of traveling wave fronts for (1). Let with . Then, (1) is transformed into the following form: where .

Before going further, we first give the definition of sharp- and smooth-type traveling wave fronts.

Definition 1. A function is called a traveling wave front with wave speed if there exist with such that is monotonic increasing and or there exist , with such that is monotonic decreasing and (i)If and , then is called an increasing sharp-type traveling wave front (see Figure 1(a)). (ii)If or , then is called an increasing smooth-type traveling wave front (see Figure 2(a)).  Similarly, we have the following.(iii)If and , then is called a decreasing sharp-type traveling wave front (see Figure 1(b)).(iv)If or , then is called a decreasing smooth-type traveling wave front (see Figure 2(b)).

fig1
Figure 1: Sharp-type traveling wave fronts. (a) Monotonic increasing. (b) Monotonic decreasing.
fig2
Figure 2: Smooth-type traveling wave fronts. (a) Monotonic increasing. (b) Monotonic decreasing.

Let . Then, (8) or (10) is transformed into Furthermore, by (9) or (11), we give the asymptotic boundary conditions for traveling wave fronts as follows: or If is strictly positive or negative for , then (12) is equivalent to Clearly, for any given , if then can be defined by However, if for some , then may be less than when is near . Therefore, the previous definition is not reasonable. In what follows, we give the definition of .

(i) If is positive, define by

  where is a solution of the following problem: (ii) If is negative, define by

  where is a solution of the following problem: Consider the following problem: In Sections 2 and 3, we will verify the following two conclusions are equivalent, that is, (1) is a monotonic solution of the problem (8)-(9) (or (10)-(11)); (2) (or ) is a solution of the problem (23).

2. Existence of Increasing Traveling Waves

In this section, we aim to find a solution of the problem (23) with for .

Since , we see that is increasing in , and so . Thus, to investigate the behavior of the trajectories of (23), we have to study the trajectories starting from , since the property of at depends on the behavior of at closely. Consider the following problem: where with is the maximal existence interval of the solution . By (24), This excludes if . Thus, we only need to find the increasing traveling wave fronts for the case . We first prove the following two lemmas.

Lemma 2. Assume that and , are solutions of (24) corresponding to different wave speeds , respectively, where  , for , and . Then, if   and if .

Proof. Recalling (19), we see that where . The desired conclusion follows immediately.

Lemma 3. For any given , and , let , be solutions of (24) corresponding to , , respectively. Then, for any , where is the maximal existence interval of the solution . In addition, . (See Figure 3(a).)

fig3
Figure 3: The properties of the trajectory . (a) The monotonicity of on . (b) The trajectory wanders through . (c) The trajectory intersects with for large .

Proof. We first show that for sufficiently small . The argument consists of three cases, , and .
(i) Consider the case . According to (24), we have Noticing that , we have for that Integrating from 0 to yields We further have Integrating from 0 to gives That is, Therefore, for sufficiently small .
(ii) Consider the case . Similar to (i), we have and, hence, Consider the sequences and , where and is sufficiently small. Noticing that and , by induction, we obtain For , if then Integrating from 0 to yields Thus, we have with and satisfying Letting , we obtain That is, Therefore, for sufficiently small .
(iii) Consider the case . Notice that which means that that is, Consequently, That is, Thus, we have Recalling (46), we see that which implies that On the other hand, by (49), we have and, hence, Summing up, we arrive at which implies that for sufficiently small .
We claim that for any . Suppose for contradiction that there exists such that and for . Then, which means that that is, Thus, Denote that . By Lemma 2, we have which means that , a contradiction.
In what follows, we will show that . Recalling the first equation of (24), we infer that By Lemma 2, for any , we have since . Thus, we obtain for any . Integrating the previous inequality from to for any yields Letting gives namely, . The proof is completed.

To deal with the behavior of the trajectories of the problem (24), we introduce the level set for any . Clearly, for , the level sets are exactly the trajectories of solutions to (12) or (15). Now, we define Notice that if is a solution of the system (12), we have since is increasing in . This implies that wanders through increasing level sets with increasing . See Figure 3(b). Let we know that passes through the critical point . Denote that In what follows, we will see that plays a special role for the proof of the main result. We first need a lemma as follows.

Lemma 4. The trajectory of the problem (24) must intersect with   for sufficiently large . (See Figure 3(c).)

Proof. For any , we have , and Let be the first point such that . Then, we have Since we have Thus, Denote that Then, for any , (74) does not hold and is increasing on . Therefore, must intersect with for any . The proof is completed.

Theorem 5. (i) If , then there is no nontrivial nonnegative solution for problem (23).
(ii) If , then there exists a unique , such that the problem (23) admits a nonnegative solution , and for any .

Proof. (i) The case has been discussed.
(ii) We know that for any fixed , wanders through increasing level sets strictly. Thus, intersects with a level set at most once. Let be the intersection point of with . Then, if , we have Define By Lemma 4, is well defined. In what follows, we will show that is just the desired wave speed.
We first have . Indeed, when , the first equation of (24) becomes Noticing that , we have Then, there exists such that since . The continuous dependence of on ensures that goes to zero before reaching for sufficiently small . Thus, .
In addition, by Lemma 3, we know that is decreasing on . Assume that If ; namely, , then by (76), we arrive at So, there exists , such that . By the continuous dependence of on , there is with and close to sufficiently such that , which implies that intersects with  . This contradicts the definition of . Thus, , which implies .
Moreover, by Lemma 3, for , we have . However, if , there exists with such that as . The proof is completed.

Proposition 6. is a monotone increasing sharp- or smooth-type traveling wave front of the problem (8)-(9) for some fixed , if and only if with for any is a solution of the problem (23).

Proof. The necessity is clear. Consider the sufficient one. Let for any be a solution of the problem (23), and solves Without loss of generality, let , and let be the maximal existence interval of such that . Firstly, we have Moreover, Therefore, if and , is a sharp-type traveling wave front; if or , is a smooth-type traveling wave front.

Theorem 5 and Proposition 6 imply the following result.

Theorem 7. (i) If , then there is no increasing traveling wave front for the problem (8)-(9).
(ii) If , then there is a unique wave speed , such that the problem (8)-(9) admits an increasing traveling wave front.

Furthermore, we have the following results.

Theorem 8. Let be the traveling wave front of the problem (8)-(9) corresponding to the wave speed .(i)If , then is a smooth-type traveling wave front. (ii)If , then and is a sharp-type traveling wave front. (iii)If , then and is a sharp-type traveling wave front.

Proof. (i) If , it is easy to see that If , from the proof of Lemma 3, we see that when is sufficiently small, Since we have Thus,
(ii) If , we have
(iii) If , we have The proof is completed.

Theorem 9. is nonincreasing in delay ; namely, if , then .

Proof. Suppose for contradiction that there exist time delays and , with and . Denote that and   for simplicity. Take such that In what follows, denote the solutions of (24) corresponding to wave speed and time delay , , by , , , respectively.
Similar to the proof of Lemma 3, we know that when is sufficiently small, for , for , for , Since , we have for sufficiently small . Furthermore, we claim that for . Otherwise, there exists such that for , and . Then, we have that is, which implies . On the other hand, by Lemma 2, we have , a contradiction. Similarly, we can get . From the uniqueness of wave speed on any fixed , this contradicts the definition of .

3. Existence of Decreasing Traveling Waves

In this section, we aim to find a solution of the problem (23) with for . We first introduce a comparison lemma.

Lemma 10. Let be the solutions of the following problems, respectively: And let solve the problem (23). Then, for if , , are positive, while for if , , are negative.

Proof. Notice that when is positive, and thus that is, where Consequently, Integrating from to 1 yields and so if . Similarly, if .
The proof for is similar and omitted here.

Since , we see that is decreasing in , and so . To get the behavior of the trajectories of (23), we have to study the trajectories starting from , since the property of at depends on the behavior of at closely. Consider the following problem: where with is the maximal existence interval of the solution . By (106), Hence, is impossible if . So, we only need to find the decreasing traveling wave fronts for the case . We first give the following three lemmas.

Lemma 11. Assume that   and  , are solutions of (106) corresponding to different wave speeds , , respectively, where  , for , and . Then, if , and if .

Proof. The proof is similar to that of Lemma 2.

Lemma 12. For any given , and , let , be solutions of (106) corresponding to , , respectively. Then, for any , where is the maximal existence interval of the solution . In addition, . (See Figure 4(a).)

fig4
Figure 4: The properties of the trajectory . (a) The monotonicity of on . (b) The trajectory wanders through . (c) The trajectory intersects with for large .

Proof. We first show that if is in a sufficiently small left neighborhood of . The argument consists of three cases, , , and .
(i) Consider the case . According to (106), we have Noticing that , we have for that Integrating from to 1 yields We further have Integrating from to 1 gives Thus, Therefore, in a left neighborhood of .
(ii)   When , consider the following two systems: Notice that the right hand side of the previous two systems shares the same Jacobian matrix at . By a simple calculation, we get the eigenvalues of the matrix as It is easy to see that is a saddle point. And the eigenvector associated with the eigenvalue can be We express the local explicit solutions of the problem (114) in the left neighborhood of to reach Therefore, near , we have By the comparison lemma, , which implies Thus, in a left neighborhood of .
(iii) Consider the case . Notice that which means that that is, Consequently, That is, Thus, we have Recalling (123), we see that which implies that