SánchezGarduño, F. and Maini, P. K. (1994) Travelling wave phenomena in some degenerate reactiondiffusion equations. Journal of Differential Equations, 114 (2). pp. 434475.

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Abstract
In this paper we study the existence of travelling wave solutions (t.w.s.), for the equation
where the reactive part g(u) is as in the FisherKPP equation and different assumptions are made on the nonlinear diffusion term D(u). Both functions D and g are defined on the interval [0, 1]. The existence problem is analysed in the following two cases.
Case 1. D(0)=0, D(u)>0 , D and , and . We prove that if there exists a value of c, c*, for which the equation (*) possesses a travelling wave solution of sharp type, it must be unique. By using some continuity arguments we show that: for 0<c<c*, there are no t.w.s., while for c>c*, the equation (*) has a continuum of t.w.s. of front type. The proof of uniqueness uses a monotonicity property of the solutions of a system of ordinary differential equations, which is also proved.
Case 2. , D and , . If, in addition, we impose with , We give sufficient conditions on c for the existence of t.w.s. of front type. Meanwhile if with we analyse just one example (, and ) which has oscillatory t.w.s. for and t.w.s. of front type for c>2. In both the above cases we use higher order terms in the Taylor series and the Centre Manifold Theorem in order to get the local behaviour around a nonhyperbolic point of codimension one in the phase plane.
Item Type:  Article 

Uncontrolled Keywords:  n/a 
Subjects:  A  C > Biology and other natural sciences 
Research Groups:  Centre for Mathematical Biology 
ID Code:  497 
Deposited By:  Philip Maini 
Deposited On:  12 Dec 2006 
Last Modified:  29 May 2015 18:23 
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