The Balance between Nonlinear Inwards and Outwards Boundary Flux for a Nonlinear Heat Equation

We study nonnegative solutions of where m, p, q, λ and L are positive parameters. For different values of the parameters three situations may occur: (1) all solutions of this problem exist for all t>0; (2) for certain initial data functions the solution exists for all t>0 while for others the solution blows up as t T for some finite T; (3) excepting the trivial solution when u0≡0, all solutions blow up as t T for some finite T. We identify in terms of the parameters which of them actually happens. For solutions which blow up we find the blow-up rate and the blow-up set.