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Zbl 1152.35323
Petitta, Francesco
Asymptotic behavior of solutions for parabolic operators of Leray-Lions type and measure data. (English)
[J] Adv. Differ. Equ. 12, No. 8, 867-891 (2007). ISSN 1079-9389

Summary: Let $\Omega\subseteq\bbfR^N$ be a bounded open set, $N\ge 2$, and let $p> 1$; we study the asymptotic behavior with respect to the time variable $t$ of the entropy solution of nonlinear parabolic problems whose model is $$\align u_t(x,t)- \Delta_p u(x,t)= \mu\quad &\text{in }\Omega\times (0,T),\\ u(x,0)= u_0(x)\quad &\text{in }\Omega,\endalign$$ where $T> 0$ is any positive constant, $u_0\in L^1(\Omega)$ a nonnegative function, and $\mu\in{\Cal M}_0(Q)$ is a nonnegative measure with bounded variation over $Q= \Omega\times(0, T)$ 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.

MSC 2000:: *35B40 Asymptotic behavior of solutions of PDE
35K55 Nonlinear parabolic equations
35R05 PDE with discontinuous coefficients or data

Keywords: entropy solution

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Zentralblatt MATH Berlin [Germany]