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

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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.

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Correspondence to Victor A. Galaktionov.

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This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.

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Galaktionov, V.A., Levine, H.A. On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary. Israel J. Math. 94, 125–146 (1996). https://doi.org/10.1007/BF02762700

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