Continuability in time of smooth solutions of strong-nonlinear nondiagonal parabolic systems

Arkhipova, Arina

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 1, pp. 153-167

Résumé

A class of quasilinear parabolic systems with quadratic nonlinearities in the gradient is considered. It is assumed that the elliptic operator of a system has variational structure. In the multidimensional case, the behavior of solutions of the Cauchy-Dirichlet problem smooth on a time interval $[0,T)$ is studied. Smooth extendibility of the solution up to $t=T$ is proved, provided that “normilized local energies” of the solution are uniformly bounded on $[0,T)$. For the case where $[0,T)$ determines the maximal interval of existence of a smooth solution,the Hausdorff measure of a singular set at the moment $t=T$ is estimated.

Détail

Classification : 35K50, 35K45, 35K60