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