On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

Abstract

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu t=uxx, ut=(um)xxand $$u_t = (\left| {u_x } \right|^{m - 1} u_x )_x $$ (m>1) forx>0,t>0 with nonlinear boundary conditions−u x=up,−(u m)x=upand $$ - \left| {u_x } \right|^{m - 1} u_x = u^p $$ forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp 0,pc(withp 0<pc)such that forp∃(0,p 0],all solutions are global while forp∃(p0,pc] any solutionu≢0 blows up in a finite time and forp>p csmall data solutions exist globally in time while large data solutions are nonglobal. We havep c=2,p c=m+1 andp c=2m for each problem, whilep 0=1,p 0=1/2(m+1) andp 0=2m/(m+1) respectively.