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# Long-time solutions of scalar nonlinear hyperbolic reaction equations incorporating relaxation I. The reaction function is a bistable cubic polynomial

We consider an initial-value problem based on a class of scalar nonlinear hyperbolic reaction–diffusion equations of the general form

𝓊_{τ τ} + 𝓊_{τ} = 𝓊_{𝓍𝓍} + *ɛ*(*F*(𝓊) + *F*(𝓊)_{τ}),

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,

*F*(𝓊) = 𝓊(1 - 𝓊)(𝓊 - *μ*),

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.

## History

## Publication title

Journal of Differential Equations## Volume

266## Issue

2-3## Pagination

1285-1312## ISSN

0022-0396## Department/School

School of Natural Sciences## Publisher

Academic Press Inc Elsevier Science## Place of publication

525 B St, Ste 1900, San Diego, USA, Ca, 92101-4495## Rights statement

Copyright 20198 Elsevier Inc.## Repository Status

- Restricted