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Blow-up rate of positive solution of uniformly parabolic equations with nonlinear boundary conditions. (English) Zbl 1047.35069

Summary: This paper deals with the blow-up of positive solutions of the uniformly parabolic equations \(u_t= Lu+ a(x) f(u)\) subject to nonlinear Neumann boundary conditions \({\partial u\over\partial n}+ b(x)g(u)= 0\). Under suitable assumptions on nonlinear functions \(f\), \(g\) and initial data \(u_0(x)\), the blow-up of the solutions in a finite time is proved by the maximum principles. Moreover, the bounds of “blow-up time” and blow-up rate are obtained.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35K55 Nonlinear parabolic equations
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
35B40 Asymptotic behavior of solutions to PDEs
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