On blow-up and dynamics near ground states for some semilinear equations

Collot, Charles

Séminaire Laurent Schwartz — EDP et applications (2015-2016), Exposé no. 20, 12 p.

The main issue of this text is the study of qualitative properties for two canonical semilinear equations:

$${\partial }_{t}u=\Delta u+{\left|u\right|}^{p-1}u,{\partial }_{tt}u=\Delta u+{\left|u\right|}^{p-1}u.$$

Its aim is threefold: introduce nonspecialists to the blow-up issue and dynamics near stationary states sometimes leading to their concentration, give an up to date bibliography on this subject for the two equations and on the existence and properties of stationary states and backward self-similar solutions, and a presentation of the author’s work on this issue with a sketch of proof for a recent result obtained in collaboration with F. Merle and P. Raphaël.

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