Electron. J. Diff. Eqns., Vol. 2008(2008), No. 132, pp. 1-10.

Large time behavior for solutions of nonlinear parabolic problems with sign-changing measure data

Francesco Petitta

Abstract:
Let $\Omega\subseteq \mathbb{R}^N$ a bounded open set, $N\geq 2$, and let $p>1$; in this paper we study the asymptotic behavior with respect to the time variable $t$ of the entropy solution of nonlinear parabolic problems whose model is
$$\displaylines{ u_{t}(x,t)-\Delta_{p} u(x,t)=\mu \quad \hbox{in } \Omega\times(0,\infty),\cr u(x,0)=u_{0}(x) \quad \hbox{in } \Omega, }$$
where $u_0 \in L^{1}(\Omega)$, and $\mu\in \mathcal{M}_{0}(Q)$ is a measure with bounded variation over $Q=\Omega\times(0,\infty)$ which does not charge the sets of zero $p$-capacity; moreover we consider $\mu$ that does not depend on time. In particular, we prove that solutions of such problems converge to stationary solutions.

Submitted June 13, 2008. Published September 23, 2008.
Math Subject Classifications: 35B40, 35K55.
Key Words: Asymptotic behavior; nonlinear parabolic equations; measure data.

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Francesco Petitta
CMA, University of Oslo,
P.O. Box 1053 Blindern, NO-0316 Oslo, Norway
email: francesco.petitta@cma.uio.no

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