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Travelling waves and finite propagation in a reaction-diffusion equation

Authors
Journal
Journal of Differential Equations
0022-0396
Publisher
Elsevier
Publication Date
Volume
93
Issue
1
Identifiers
DOI: 10.1016/0022-0396(91)90021-z

Abstract

Abstract We study the existence of travelling wave solutions and the property of finite propagation for the reaction-diffusion equation u t = ( u m ) xx + λu n , ( x, t) ϵ R × (0, ∞) with m > 1, λ > 0, n ϵ R, and u = u( x, t) ⩾ 0. We show that travelling waves exist globally only if m + n = 2 and only for velocities ¦c¦ ⩾ c ∗ = 2 √λm . In the study of propagation we must take into account that solutions of the Cauchy problem can be nonunique for n < 1. Finite propagation occurs for minimal solutions if and only if m + n ⩾ 2, and there exists a minimal velocity c ∗ > 0 for m + n = 2. Maximal solutions propagate instantly to the whole space if n < 1.

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