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Zbl 1017.35040
Droniou, Jérôme; Porretta, Alessio; Prignet, Alain
Parabolic capacity and soft measures for nonlinear equations.
(English)
[J] Potential Anal. 19, No.2, 99-161 (2003). ISSN 0926-2601; ISSN 1572-929X/e

Let $\Omega$ be a bounded, open subset of $\bbfR^N$, $T$ a positive number and $Q=[0,T]\times \Omega$. Let $p$ be a real number, with $1<p<\infty$, and let $1/p+1/p'=1$. This paper deals with the parabolic initial-boundary value problem $u_t+A(u)=\mu$ in $Q$, $u=0$ on $[0,T]\times \partial \Omega$, $u(0)=u_0$ in $\Omega$, where $A$ is a nonlinear monotone and coercive operator in divergence form which acts from the space $L^p(0,T;W_0^{1,p}(\Omega))$ into its dual $L^{p'}(0,T; W^{-1,p'}(\Omega))$. The authors first introduced the notion of capacity related to the parabolic $p$-Laplace operator. Then they proved a decomposition theorem for measures that do not charge the sets with null capacity. Using these results they proved the following Theorem: Let $\mu$ be a bounded real measure on $Q$ which does not charge the subsets of $Q$ of null capacity, and let $u_0 \in L^1(\Omega)$. Then there is a unique renormalized solution $u$ of the problem. Moreover, $u$ satisfies the additional regularity: $u\in L^\infty (0,T; L^1(\Omega))$ and $\max (-k,\min (k,u)) \in L^p(0,T;W_0^{1,p}(\Omega))$ for every $k>0$.
[Dagmar Medková (Praha)]
MSC 2000:
*35K55 Nonlinear parabolic equations

Keywords: parabolic equation; capacity

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