Leach, JA and Bassom, AP, Long-time solutions of scalar nonlinear hyperbolic reaction equations incorporating relaxation I. The reaction function is a bistable cubic polynomial, Journal of Differential Equations, 266, (2-3) pp. 1285-1312. ISSN 0022-0396 (2019) [Refereed Article]
Copyright 20198 Elsevier Inc.
We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general form
in which 𝓍 and τ represent dimensionless distance and time respectively and ɛ > 0 is a parameter related to the relaxation time. Furthermore the reaction function, F(𝓊), is given by the bistable cubic polynomial,
in which 0 < μ < 1—2 is a parameter. The initial data is given by a simple step function with 𝓊(𝓍, 0) = 1 for 𝓍 ≤ 0 and 𝓊(𝓍, 0) for 𝓍 > 0. It is established, via the method of matched asymptotic expansions, that the large-time structure of the solution to the initial-value problem involves the evolution of a propagating wave front which is either of reaction–diffusion or of reaction–relaxation type. The one exception to this occurs when μ = ½ in which case the large time attractor for the solution of the initial-value problem is a stationary state solution of kink type centred at the origin.
|Item Type:||Refereed Article|
|Keywords:||differential equations, asymptotics|
|Research Division:||Mathematical Sciences|
|Research Group:||Applied Mathematics|
|Research Field:||Theoretical and Applied Mechanics|
|Objective Division:||Expanding Knowledge|
|Objective Group:||Expanding Knowledge|
|Objective Field:||Expanding Knowledge in the Mathematical Sciences|
|UTAS Author:||Bassom, AP (Professor Andrew Bassom)|
|Deposited By:||Mathematics and Physics|
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