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On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary. (English) Zbl 0851.35067
Summary: We consider nonnegative solutions of initial-boundary value problems for parabolic equations $$u_t= u_{xx}$$, $$u_t= (u^m)_{xx}$$ and $$u_t= (|u_x|^{m- 1} u_x)_x$$ $$(m> 1)$$ for $$x> 0$$, $$t> 0$$ with nonlinear boundary conditions $$- u_x= u^p$$, $$-(u^m)_x= u^p$$ and $$- |u_x|^{m- 1} u_x= u^p$$ 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 $$p\in (0, p_0]$$, all solutions are global while for $$p\in (p_0, p_c]$$ any solution $$a\not\equiv 0$$ 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 $$p_0= 1$$, $$p_0= {1\over 2} (m+ 1)$$ and $$p_0= 2m(m+ 1)$$ respectively.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
critical Fujita exponents; blow up
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##### References:
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