Israel Journal of Mathematics

, Volume 94, Issue 1, pp 125–146

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

Article

DOI: 10.1007/BF02762700

Cite this article as:
Galaktionov, V.A. & Levine, H.A. Israel J. Math. (1996) 94: 125. doi:10.1007/BF02762700

Abstract

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsut=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−ux=up,−(um)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 valuesp0,pc(withp0<pc)such that forp∃(0,p0],all solutions are global while forp∃(p0,pc] any solutionu≢0 blows up in a finite time and forp>pcsmall data solutions exist globally in time while large data solutions are nonglobal. We havepc=2,pc=m+1 andpc=2m for each problem, whilep0=1,p0=1/2(m+1) andp0=2m/(m+1) respectively.

Copyright information

© The Magnes Press 1996

Authors and Affiliations

  1. 1.Keldysh Institute of Applied MathematicsRussian Acad. Sci.MoscowRussia
  2. 2.Department of MathematicsIowa State UniversityAmesU.S.A.
  3. 3.Departamento de MatematicasUniversidad Autonoma de MadridMadridSpain