A quasi-monotonicity formula and partial regularity for borderline solutions to a parabolic equation
Annales de l'I.H.P. Analyse non linéaire, Volume 27 (2010) no. 6, p. 1333-1360

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

DOI : https://doi.org/10.1016/j.anihpc.2010.07.001
Keywords: Quasi-monotonicity formula, Partial regularity, Borderline solutions, Semilinear parabolic equations, Potential
@article{AIHPC_2010__27_6_1333_0,
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},
publisher = {Elsevier},
volume = {27},
number = {6},
year = {2010},
pages = {1333-1360},
doi = {10.1016/j.anihpc.2010.07.001},
zbl = {1213.35177},
mrnumber = {2738324},
language = {en},
url = {http://www.numdam.org/item/AIHPC_2010__27_6_1333_0}
}

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, Volume 27 (2010) no. 6, pp. 1333-1360. doi : 10.1016/j.anihpc.2010.07.001. http://www.numdam.org/item/AIHPC_2010__27_6_1333_0/

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