Some semi-linear parabolic equations with nonlinear flux boundary conditions

We consider a general class of semi-linear parabolic equations defined on simple ball shaped domains in n dimensions, that are subject to nonlinear boundary conditions. Domain size plays the role of a natural bifurcation parameter balancing the strength of the internal source term over the domain against the nonlinear flux term active at the boundary. When the source term is linear, then using spectral theory arguments we show how a simple condition on the nonlinearity within the boundary condition completely determines the stability question for radially symmetric steady state solutions. For super-linear sources we must resort to using the comparison principle argument that yields an identical condition sufficient for their instability, but fails to provide the converse. We show that when applied to a class of (hemispherical asymmetric) symmetry breaking solutions, available for the corresponding steady state elliptic equation, these cannot be stable as solutions for the parabolic system. Hence such symmetry breaking may lead to large instabilities. We illustrate the results with some simple examples.