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

and Applied Analysis 3 (a) (b) Figure 1: Sharp-type traveling wave fronts. (a) Monotonic increasing. (b) Monotonic decreasing. (a) (b) Figure 2: Smooth-type traveling wave fronts. (a) Monotonic increasing. (b) Monotonic decreasing. Clearly, for any given φ > 0, if

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, [2][3][4][5][6][7].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, [8][9][10].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  > 1, the equation degenerates at  = 0. Hence, it has a different feature from the case  = 1; that is, if the initial distribution of (, ) has compact support, then (, ) also has compact support for each  > 0. When  > 1,  ∈ (0, 1/2), 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: ∈ (0, 1) ,  (0) = 0, (5) 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 [14][15][16][17][18][19][20][21][22].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 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.
Before going further, we first give the definition of sharpand smooth-type traveling wave fronts.Definition 1.A function () is called a traveling wave front with wave speed  > 0 if there exist   ,   with −∞ ≤   <   ≤ +∞ such that  ∈  2 (  ,   ) is monotonic increasing and or there exist ξ , ξ with −∞ ≤ ξ < ξ ≤ +∞ such that  ∈  2 ( ξ , ξ ) is monotonic decreasing and is called an increasing sharp-type traveling wave front (see Figure 1(a)).
(ii) If   = −∞ or   ( +  ) = 0, then () is called an increasing smooth-type traveling wave front (see Figure 2 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 0 <  < 1, then ( 12) is equivalent to Clearly, for any given  > 0, if then   can be defined by However, if for some  > 0, then ∫  0 ( −1 /()) may be less than  when  is near 0. 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: 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) () > 0 (or () < 0) is a solution of the problem (23).

Existence of Increasing Traveling Waves
In this section, we aim to find a solution () of the problem (23) with () > 0 for  ∈ (0, 1).
(i) Consider the case  +  > 2. 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,  1 () >  2 () for sufficiently small  > 0.
(iii) Consider the case 1 <  +  < 2. 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  1 () >  2 () for sufficiently small  > 0.
To deal with the behavior of the trajectories   () of the problem (24), we introduce the level set for any  ≥ 0. Clearly, for  = 0, 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   1 passes through the critical point (1, 0).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.Proof.For any  ∈ (0, ], we have   ∈ (0, ], and Let  0 ∈ (, 1) be the first point such that    ( 0 ) = 0.Then, we have Since we have Thus, Denote that Then, for any  ≥ √ , (74) does not hold and   () is increasing on (, 1).Therefore,   () must intersect with  * () for any  ≥ √ .The proof is completed.
then there is no nontrivial nonnegative solution for problem (23).
Theorem 5 and Proposition 6 imply the following result.
then there is no increasing traveling wave front for the problem (8)-( 9). (ii then there is a unique wave speed  * 1 > 0, such that the problem (8)-( 9) admits an increasing traveling wave front.
Furthermore, we have the following results.Proof.(i) If 0 <  ≤ 1, it is easy to see that If 1 <  < 2, from the proof of Lemma 3, we see that when  > 0 is sufficiently small,  () =  * 1  +  () ,  +  > 2; Since we have Thus, (ii) If  = 2, we have (iii) If  > 2, we have The proof is completed.

Existence of Decreasing Traveling Waves
In this section, we aim to find a solution () of the problem (23) with () < 0 for  ∈ (0, 1).We first introduce a comparison lemma.
The proof for , , V < 0 is similar and omitted here.
(i) Consider the case  > 1.According to (106), we have Noticing that   ≥ , we have for  ≥  that Integrating from  to 1 yields We further have Integrating from  to 1 gives (ii) When  = 1, consider the following two systems: Notice that the right hand side of the previous two systems shares the same Jacobian matrix at (1, 0).By a simple calculation, we get the eigenvalues of the matrix  as It is easy to see that (1, 0) 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 (1, 0) to reach Therefore, near (1, 0), we have By the comparison lemma,  2 ≤  ≤  1 , which implies Thus,  1 () <  2 () in a left neighborhood of  = 1.
(iii) Consider the case 0 <  < 1.Notice that which means that that is, Consequently, That is, Thus, we have Recalling (123), we see that which implies that On the other hand, by (126), we have and, hence, Summing up, we arrive at which implies that  1 () <  2 () in a left neighborhood of  = 1.
Similar to Section 2, we denote level set   by for any  ≥ 0, and correspondingly, define Notice that if (, ) ∈  −  solves system (12), then since () is decreasing in .This implies that the trajectory (, ) of ( 106) wanders through increasing level sets with increasing .See Figure 4(b).Letting we know that   2 passes through the critical point (0, 0).Denote that We introduce the following lemma.
then there is no nontrivial nonpositive solution for the problem (23).
Proof.The proof is similar to that of Theorem 5, and Proposition 15. () is a monotone decreasing sharp-or smooth-type traveling wave front of the problem (10)-( 11) for some fixed  > 0, if and only if () with () < 0 for any  ∈ (0, 1) is a solution of the problem (23).
Proof.The proof is similar to that of Proposition 6.
Theorem 14 and Proposition 15 imply the following result.
then there is no decreasing traveling wave front for the problem (10)- (11).
(ii) If ∫ 1 0  +−1 (1 − )  ( − ) < 0, then there is a unique wave speed  * 2 > 0, such that the problem (10)-( 11) admits a decreasing traveling wave front.Furthermore, we have the following results. Thus, Recalling that we know that which means that The proof is completed.
Proof.The proof is similar to that of Theorem 9.
(ii) It is easy to see that Thus, Letting  → 1, we obtain that is, For any  > 0, when  approaches 1 enough, we have   * 1  > 1 − .Consider the following two problems: for  ≤ 1, It is easy to prove that V() ≤ () ≤ () when  < 1 with In what follows, we construct two functions  and V satisfying (162) and (163), respectively.Let Then,  satisfies (162) if and only if When  > 1, take When  = 1, take Then,  is a solution of problem (162).
(ii) From the proof of Theorem 14, we know that On the other hand, from the proof of Theorem 17, we also note that By ( 218)-(219) and it is easy to see that if  ≥ min{, 1}.That is, For any 0 <  < , when  approaches 0 enough, we have   * 2  < .Consider the following problem:

Discussion
When  =  = 1, the outcome in our work is reduced to the results obtained in [23].Comparing with [23], our definition of sharp-type traveling wave fronts and smooth-type traveling wave fronts is more precise.In the proof of Lemma 3, for the case  +  = 2, we constructed two sequences to get the asymptotic expression of () for  > 0 sufficiently small.This technique is not used in [23].Moreover, the proof of Theorem 17 is more concise than the proof of Proposition 2.7 in [23].

Theorem 17 .
The traveling wave front () of the problem (10)-(11) corresponding to the wave speed  * 2 obtained earlier is of smooth type.Proof.It is easy to see that   ≤ 0, as  → 0 + .