Existence and Uniqueness of Blow-up Solutions for a Parabolic Problem with a Localized Nonlinear Term via Semi-group Theory

Panumart Sawangtong (1) , Christian Licht (2) , Boriboon Novaprateep (3) , Somsak Orankitjaroen (4)

1 Department of Mathematics
2 M3 - Modélisation Mathématique en Mécanique
3 Department of Mathematics
4 Department of Mathematics

Here, we use the semigroup theory to establish the existence, uniqueness and blow-up for a classical solution of a semilinear parabolic problem with localized nonlinear term| a locally Lipschitz continuous function of the value of the solution at a point of a 1-dimensional domain. Our method, which uses Sobolev spaces and fractional power of operators, is in contrast with the classical ones (Green functions) which supply similar results in 1-dimensional settings.

Mots clés: semilinear parabolic problems; blow-up; semigroups of linear operator

Domaines: Matériaux et structures en mécanique [physics.class-ph]; Matériaux et structures en mécanique [physics.class-ph]; Mécanique des matériaux [physics.class-ph]; Mécanique des matériaux [physics.class-ph]; Mécanique des solides [physics.class-ph]; Mécanique des solides [physics.class-ph]; Mécanique des structures [physics.class-ph]; Mécanique des structures [physics.class-ph]

Panumart Sawangtong, Christian Licht, Boriboon Novaprateep, Somsak Orankitjaroen. Existence and Uniqueness of Blow-up Solutions for a Parabolic Problem with a Localized Nonlinear Term via Semi-group Theory. International Conference on Mathematics and Applications, ICMA-MU 2009, Dec 2009, Thailand. pp.139-152. ⟨hal-00747045⟩