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Parabolic capacity and soft measures for nonlinear equations. (English) Zbl 1017.35040
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 $k>0$.

##### MSC:
 35K55 Nonlinear parabolic equations
##### Keywords:
parabolic equation; capacity
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