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On a nonlinear heat equation with a time delay

Published online by Cambridge University Press:  16 July 2002

YUNKANG LIU
YUNKANG LIU
Affiliation:
Department of Mathematical Sciences, Loughborough University, Loughborough, Leicestershire LE11 3TU, UK email: [email protected] Current address: 34 Cedar Close, Buckhurst Hill, Essex IG9 6EJ, UK.
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Abstract

A nonlinear forward-backward heat equation with a regularization term was proposed by Barenblatt et al. [1, 2] to model the heat and mass exchange in stably stratified turbulent shear flow. It was proven to be well-posed in the case of given initial and Neumann boundary conditions. However, the solution was found to have an unphysical discontinuity with certain smooth initial functions. In this paper, a nonlinear heat equation with a time delay originally used by Barenblatt et al. [1, 2] to derive their model is investigated. The same type of initial-boundary value problem is shown to have a unique smooth global solution when the initial function is reasonably smooth. Numerical examples are used to demonstrate that its solution forms step-like profiles in finite times. A semi-discretization of the initial-boundary value problem is proved to have a unique asymptotically and globally stable equilibrium.

Type
Research Article
Information
European Journal of Applied Mathematics , Volume 13 , Issue 3 , June 2002 , pp. 321 - 335
Copyright
2002 Cambridge University Press

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