Exact solutions of hyperbolic reaction-diffusion equations in two dimensions





reaction-diffusion, hyperbolic diffusion, population dynamics, combustion


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 \(\tau\).


doi: 10.1017/S1446181123000093

Author Biographies

Phil Broadbridge, La Trobe University

Department of Mathematical and Physical Sciences, La Trobe University, Bundoora VIC 3086, Australia

Joanna Goard, University of Wollongong

School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia





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