Global higher regularity of solutions to singular p(x, t)-parabolic equations

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:466

Article

Antontsev, Stanislav^1,2,3  Kuznetsov, Ivan^2,3  Shmarev, Sergey⁴ 

[1] Univ Lisbon, CMAF CIO, Lisbon, Portugal

[2] RAS, Lavrentyev Inst Hydrodynam, SB, Novosibirsk, Russia

[3] Novosibirsk State Univ, Novosibirsk, Russia

[4] Univ Oviedo, Dept Math, Oviedo, Spain

关键词: Singular parabolic equation;    Variable nonlinearity;    Higher regularity;    Strong solutions;   

DOI  :  10.1016/j.jmaa.2018.05.075

来源: Elsevier

【 摘 要 】

We study the homogeneous Dirichlet problem for the equation u(t) = div(vertical bar del u vertical bar(p(x, t)-2)del u) + f (x, t, u) in the cylinder Q(T) = Omega x (0, T), Omega subset of R-d, d >= 2. It is assumed that p(x, t) is an element of (2d/d+2, 2) and vertical bar del p vertical bar, vertical bar pt vertical bar are bounded a.e. in Q(T). We find conditions on p(x,t), f(x,t,u) and u(x, 0) sufficient for the existence of strong solutions, local or global in time. It is proven that the strong solutions possess the property of global higher regularity: u(t) is an element of L-2 (Q(T)), vertical bar del u vertical bar is an element of L-infinity (0, T; L-2 (Omega)), vertical bar D(ij)(2)u vertical bar(p(x, t)) is an element of L-1 (QT). (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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