Transient behavior of solutions to a class of nonlinear boundary value problems
نویسندگان
چکیده
In this paper we consider the asymptotic behavior in time of solutions to the heat equation with certain nonlinear Neumann boundary conditions, ∂u/∂n = F (u). Here F is a function which grows superlinearly. In general solutions exist for only a finite time before “blowing up”, or they decay to zero as time approaches infinity. In both one and two space-dimensions we establish some conditions on the initial data u(·, 0) under which blow-up is assured, and other conditions that lead to decay. In one spacedimension we perform a detailed examination of the nature of the blow-up, which can occur only at the boundary, and provide tight lower and upper bounds on the blow-up rate for “arbitrary” nonlinear functions F , subject to mild restrictions.
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