Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 132, pp. 1-10.
Title: Large time behavior for solutions of nonlinear parabolic problems
with sign-changing measure data
Author: Francesco Petitta (Univ. of Oslo, Norway)
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.