Traveling Wavefronts on Reaction Diffusion Systems with Spatio-Temporal Delays

By using Schauder’s Fixed Point Theorem, we study the existence of traveling wave fronts for reaction-diffusion systems with spatio-temporal delays. In our results, we reduce the existence of traveling wave fronts to the existence of an admissible pair of upper solution and lower solution which are much easier to construct in practice.


Introduction
Traveling wave solutions, usually characterized as solutions invariant with respect to translation in space, have attracted much attention due to their significant nature in science and engineering [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].In which, the theory of wave fronts of reaction diffusion systems is an important part, and its history traces back to the so-called Fisher-KPP equation, the celebrated mathematical works by P. A. Fisher and by Kolmogorov, Petrovskii and Piscunov.Since then, lots of papers are devoted to the study of traveling wave solutions of reaction diffusion systems, and various research methods come forth.
The present paper is mainly devoted to tackle the existence of traveling wave front solutions of the following reaction diffusion system with spatial-temporal delays and with some zero-diffusive coefficients, , , , , ( And the kernels used frequently in the reference are as follows 1)       , ;       , ; The remaining part of this paper is organized as follows.In the next section, some preliminaries are given.In Section 3, we state and prove the main result of this paper.

Preliminaries
A traveling wave solution of (1.1) is a special translation invariant solution of the form is the profile of the wave that propagates through the one-dimensional spatial domain at a constant velocity c > 0. If is monotone and satis- fies the asymptotic boundary conditions   lim s s U  wave front of (1.1), where  , and U − , U + are equilibria of system (1.1)   where  will be given in the next section.Obviously, , and are Banach spaces respectively with the norms : sup e , ; , Substituting into (1.1) and denoting still by t the traveling coordinate x ct  , we obtain the corresponding wave equations where c > 0 is velocity, , , Without loss of generality, we assume In the following, we list the basic assumptions of this paper: (A 1 ) .

   
, , , , 0 There exists an constant 0   such that for and 1, , , One of the following two cases holds.

 
1 diag , , , 0,   where , and n in 2 ).In dition, we can obtain We only give the proof under th ase an e c (A ) is similar.Firstly, we show For fixed let , then by Hence , there is a constant where   1 min .
By , for the above A, there for the above constan  ,  , m and L, th ere exists an 3. en by (3.2)-( 4) and (A 2 ), we have .

s y t y cs V s y g s y t y cs V s y g s y t y cs V s y
, , .
lim lim inf and and 0.

 
H  is also nondecreasi Proof. 1) and 2) can be given directly by (A 1 ) and (A 5 ).
3) For ng in .
By the monotonicity of and this complete the proof of Lemma 3.1.Without loss of generality, we assume 0 i   in (A 5 ), and denote Defining the integral operator P on , by ma 3.1 1 Lem ), we have on, similar to the pr Therefore, In additi ˆôof of Proposition 3.2 we can obtain, 0 n B > 0 And for this A, there is a consta t s.t. for

  
Obviously, * then by (A 2 ) we obtain In addition, by calculating directly we can obtain the following, if . 0, On the other hand, by the a So we have where  is a constant. .
e is ilar to the method in Ma [4], we obtain ther G > 0 s.t.
, we ow P is continuous with respect tothe norm By the continuity of H kn u  .The proof is completed.In the following, we state and prove the main theorem of this paper.
 we first prove two lemmas.hold, then for , , then we have   ; n W BC    .In order to obtain From the definition of lower solution, we know and by Lemma 3.1 2), we obtain Considering By the continuity of and formula of va onstants, we have where  is a c . and, This can be easily   pr lowed by .4. oved fol Proposition 3.3 and Lemma 3.2 -3 Step 4,   ,  Step 5, (2.1) and ( 2 , g U t x g t s x y U s y y s

3 . 2 .
This completes the pro is uniformly conposition Assume (A 2 ) and (A 4 ) hold.If wave eq tinuous of of the proposition.Pro uations (2.1) have a monotone solution . If Y < Z, we also denote Let  be the supremum norm in n and 