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

## Authors

- Received:
- Revised:

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 equations*u*
_{t}=u_{xx}, u_{t}=(u^{m})_{xx}and\(u_t = (\left| {u_x } \right|^{m - 1} u_x )_x \) (*m*>1) for*x*>0,*t*>0 with nonlinear boundary conditions−*u*
_{x}=u^{p},−(*u*
^{m})_{x}=u^{p}and\( - \left| {u_x } \right|^{m - 1} u_x = u^p \) for*x*=0,*t*>0, where*p*>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 values*p*
_{0},p_{c}(with*p*
_{0}<p_{c})such that for*p*∃(0,*p*
_{0}],all solutions are global while for*p∃*(p_{0},p_{c}] any solution*u*≢0 blows up in a finite time and for*p>p*
_{c}small data solutions exist globally in time while large data solutions are nonglobal. We have*p*
_{c}=2,*p*
_{c}=m+1 and*p*
_{c}=2m for each problem, while*p*
_{0}=1,*p*
_{0}=1/2(m+1) and*p*
_{0}=2m/(m+1) respectively.