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EXACT SOLUTIONS OF HYPERBOLIC REACTION-DIFFUSION EQUATIONS IN TWO DIMENSIONS
- Part of
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- Journal:
- The ANZIAM Journal / Volume 64 / Issue 4 / October 2022
- Published online by Cambridge University Press:
- 17 July 2023, pp. 338-354
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- Article
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Exact solutions are constructed for a class of nonlinear hyperbolic reaction-diffusion equations in two-space dimensions. Reduction of variables and subsequent solutions follow from a special nonclassical symmetry that uncovers a conditionally integrable system, equivalent to the linear Helmholtz equation. The hyperbolicity is commonly associated with a speed limit due to a delay,
$\tau $, between gradients and fluxes. With lethal boundary conditions on a circular domain wherein a species population exhibits logistic growth of Fisher–KPP type with equal time lag, the critical domain size for avoidance of extinction does not depend on 
$\tau $. A diminishing exact solution within a circular domain is also constructed, when the reaction represents a weak Allee effect of Huxley type. For a combustion reaction of Arrhenius type, the only known exact solution that is finite but unbounded is extended to allow for a positive 
$\tau $.
