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

AbstractWe study nonnegative solutions ofut=(um)xx(x,t)∈(0,L)×(0,T),−(um)x(0,t)=up(0,t),t∈(0,T),(um)x(L,t)=−λuq(L,t),t∈(0,T),u(x,0)=u0(x),x∈(0,L), 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