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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.

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
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