Exact solutions for nonlinear Arrhenius reaction-diffusion are constructed in$n$ dimensions. A single relationship between nonlinear diffusivity and thenonlinear reaction term leads to a nonclassical Lie symmetry whose invariantsolutions have a heat flux that is exponential in time (either growth ordecay), and satisfying a linear Helmholtz equation in space. This constructionextends also to heterogeneous diffusion wherein the nonlinear diffusivityfactorises to the product of a function of temperature and a function ofposition. Example solutions are given with applications to heat conduction inconjunction with either exothermic or endothermic reactions, and to soil-waterflow in conjunction with water extraction by a web of plant roots.
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