EXACT SOLUTIONS OF HYPERBOLIC REACTION-DIFFUSION EQUATIONS IN TWO DIMENSIONS
Publication Name
ANZIAM Journal
Abstract
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, τ, 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 τ. 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 τ.
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