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

Zheng, Gao-Feng

doi:10.1016/j.anihpc.2010.07.001

Zheng, Gao-Feng

Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1333-1360

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Résumé

A quasi-monotonicity formula for the solution to a semilinear parabolic equation $u_{t} = Δ u + V (x) {| u |}^{p - 1} u$ , $p > (N + 2) / (N - 2)$ in $Ω \times (0, T)$ 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 $Q \subset Ω \times (0, T)$ there exists a close subset $Q^{'} \subset Q$ such that u is continuous in $Q^{'}$ and the $(N - \frac{4}{p - 1})$ -dimensional parabolic Hausdorff measure $ℋ^{(N - \frac{4}{p - 1})} (Q ∖ Q^{'})$ of $Q ∖ Q^{'}$ is finite.

MR Zbl

DOI : 10.1016/j.anihpc.2010.07.001

Keywords: Quasi-monotonicity formula, Partial regularity, Borderline solutions, Semilinear parabolic equations, Potential

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     author = {Zheng, Gao-Feng},
     title = {A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1333--1360},
     publisher = {Elsevier},
     volume = {27},
     number = {6},
     year = {2010},
     doi = {10.1016/j.anihpc.2010.07.001},
     mrnumber = {2738324},
     zbl = {1213.35177},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2010.07.001/}
}

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Zheng, Gao-Feng. A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 6, pp. 1333-1360. doi: 10.1016/j.anihpc.2010.07.001

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