nonlinear diffusion reaction equation
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2020 ◽  
Vol 81 (6-7) ◽  
pp. 1495-1522
Author(s):  
Yifei Li ◽  
Peter van Heijster ◽  
Robert Marangell ◽  
Matthew J. Simpson

AbstractWe use a geometric approach to prove the existence of smooth travelling wave solutions of a nonlinear diffusion–reaction equation with logistic kinetics and a convex nonlinear diffusivity function which changes sign twice in our domain of interest. We determine the minimum wave speed, $$c^*$$ c ∗ , and investigate its relation to the spectral stability of a desingularised linear operator associated with the travelling wave solutions.


Author(s):  
R. E. Grundy

AbstractIn this paper we consider the asymptotics of extinction for the nonlinear diffusion reaction equationwith non-negative initial data possessing finite support. For t > 0, both solution and support vanish as t → T and x → x0. With T as the extinction time we construct the asymptotic solution as τ = T – t → 0 near the extinction point x0 using matched expansions. Taking x0= 0, we first form an outer expansion valid when η =xt–(m–p)/2 (1–p) = 0(1). This is nonuniformly valid for large |η| and has to be replaced by an intermediate expansion valid for |x| = O(τ−1/l0) where l0 is an even integer greater than unity. If p + m ≥ 2 this expansion is uniformly valid while for p + m < 2, there are regions near the edge of the support where diffusion becomes important. The zero order solution in these inner regions is discussed numerically.


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