A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation

Zheng, Gao-Feng

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010), p. 1333-1360 / Harvested from Numdam

Résumé

A quasi-monotonicity formula for the solution to a semilinear parabolic equation $ {u}_{t}=\mathrm{\Delta }u+V(x){|u|}^{p-1}u$, $ p>(N+2)/(N-2)$ in $ \Omega \times (0,T)$ with 0-Dirichlet boundary condition is obtained. As an application, it is shown that for some suitable global weak solution u and any compact set $ Q\subset \Omega \times (0,T)$ there exists a close subset $ {Q}^{\prime }\subset Q$ such that u is continuous in $ {Q}^{\prime }$ and the $ (N-\frac{4}{p-1})$-dimensional parabolic Hausdorff measure $ {\mathcal{H}}^{(N-\frac{4}{p-1})}(Q\setminus {Q}^{\prime })$ of $ Q\setminus {Q}^{\prime }$ is finite.

Détail

Publié le : 2010-01-01

MR 2738324 | Zbl 1213.35177

DOI : https://doi.org/10.1016/j.anihpc.2010.07.001

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