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Energy Methods for Problems with Nonhomogeneous Boundary Conditions
Authors:F. Bofill  R. Quintanilla
Affiliation:(1) Matemática Aplicada 2, Universidad Politècnica de Catalunya, Colón, 11, Terrassa, Barcelona, Spain
Abstract:This paper employs the weighted energy method to derive estimates for the dynamic behavior of solutions to boundary and initial boundary value problems with nonhomogeneous boundary conditions. In particular, the method is applied to the heat and Laplace equations in a bounded or unbounded region. Extensions to related equations are also studied. Similar estimates but for the spatial behavior is obtained for the heat equation and the backward in time heat equation. Results for blow-up in finite time of solutions to certain nonlinear equations are generalized to include nonhomogeneous boundary conditions, while solutions that vanish on part of the boundary are briefly discussed in the final section.
Keywords:weighted energy method  decay estimates  Laplace equation  heat equation  porous medium equation  nonhomogeneous boundary conditions
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