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Complete and energy blow-up in parabolic problems with nonlinear boundary conditions

Rodríguez Bernal, Aníbal and Quittner, Pavol (2005) Complete and energy blow-up in parabolic problems with nonlinear boundary conditions. Nonlinear Analysis Theory Methods and Applications , 62 (5). 863-875 . ISSN 0362-546X

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Abstract

We study the possible continuation of solutions of a nonlinear parabolic problem after the blow-up time. The nonlinearity in the equation is dissipative and blow-up is caused by the nonlinear boundary condition of the form ∂u/∂ν=|u|q-1u, where q>1 is subcritical in H1(Ω). If the dissipative term in the equation is linear then we show that blow-up of positive solutions is complete. If the dissipative term is superlinear then the solution can be continued inside the spatial domain. On the other hand, we find sufficient conditions on the nonlinearities guaranteeing that no reasonable continuation can be expected on the boundary.


Item Type:Article
Uncontrolled Keywords:Superlinear parabolic problem; Nonlinear boundary condition; A priori estimate; Complete blow-up
Subjects:Sciences > Mathematics > Differential equations
ID Code:12770
Deposited On:27 May 2011 08:30
Last Modified:06 Feb 2014 09:32

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