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Discrete and Continuous Dynamical Systems - Series S (DCDS-S)
 

Some degenerate parabolic problems: Existence and decay properties

Pages: 617 - 629, Volume 7, Issue 4, August 2014      doi:10.3934/dcdss.2014.7.617

 
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Lucio Boccardo - Dipartimento di Matematica, Sapienza Universit√° di Roma, Piazzale A. Moro 5, 00185 Roma, Italy (email)
Maria Michaela Porzio - Dipartimento di Matematica, Sapienza Universit√° di Roma, Piazzale A. Moro 5, 00185 Roma, Italy (email)

Abstract: We study the existence of solutions $u $ belonging to $L^1(0,T; W_0^{1,1}(\Omega)) \cap L^{\infty}(0,T;L^2(\Omega))$ of a class of nonlinear problems whose prototype is the following \begin{equation} \label{prob1} \left\{ \begin{array}{lll} \displaystyle u_t - {\rm div} \left( \frac{\nabla u}{(1+|u|)^2} \right) = 0, & \hbox{in} & \Omega_T; \\ u=0, & \hbox{on} & \partial\Omega \times (0,T); & & & & \hbox{(1)}\\ u(x,0)= u_0(x) \in L^2(\Omega), & \hbox{ in} & \Omega. \end{array} \right. \end{equation} We investigate also the asymptotic estimates satisfied by distributional solutions that we find and the uniqueness.

Keywords:  Nonlinear parabolic problems, degenerate parabolic equations, $L^1(0,T;W_0^{1,1}(\Omega))$-solutions, decay estimates, unbounded solutions, existence results.
Mathematics Subject Classification:  Primary: 35K65, 35K55; Secondary: 35K10, 35K15, 35K20.

Received: October 2013;      Revised: December 2013;      Available Online: February 2014.

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