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 $.

构造了一类二维空间非线性双曲反应扩散方程的精确解。变量的约简和随后的解遵循特殊的非经典对称性，该对称性揭示了一个条件可积系统，相当于线性亥姆霍兹方程。由于梯度和通量之间的延迟$\tau $ ，双曲性通常与速度限制相关。在圆形域上的致命边界条件下，其中物种种群表现出具有相同时间滞后的 Fisher-KPP 类型的 Logistic 增长，避免灭绝的临界域大小不取决于 $\tau $。当反应表现出 Huxley 型的弱 Allee 效应时，还构建了圆形域内的递减精确解。对于阿累尼乌斯类型的燃烧反应，唯一已知的有限但无界的精确解被扩展为允许正$\tau $。