# American Institute of Mathematical Sciences

January  2007, 18(1): 15-38. doi: 10.3934/dcds.2007.18.15

## Well-posedness and long-time behavior for a class of doubly nonlinear equations

 1 Università degli Studi di Pavia, Dipartimento di Matematica "F. Casorati", Via Ferrata 1, 27100 Pavia 2 Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany 3 Istituto di Matematica Applicata e Tecnologie Informatiche – CNR, Via Ferrata 1, 27100 Pavia, Italy

Received  May 2006 Revised  December 2006 Published  February 2007

This paper addresses a doubly nonlinear parabolic inclusion of the form

$\mathcal A (u_t)+\mathcal B (u)$ ∋ f.

Existence of a solution is proved under suitable monotonicity, coercivity, and structure assumptions on the operators $\mathcal A$ and $\mathcal B$, which in particular are both supposed to be subdifferentials of functionals on $L^2(\Omega)$. Since unbounded operators $\mathcal A$ are included in the analysis, this theory partly extends Colli & Visintin's work [24]. Moreover, under additional hypotheses on $\mathcal B$, uniqueness of the solution is proved. Finally, a characterization of $\omega$-limit sets of solutions is given, and we investigate the convergence of trajectories to limit points.

Citation: Giulio Schimperna, Antonio Segatti, Ulisse Stefanelli. Well-posedness and long-time behavior for a class of doubly nonlinear equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (1) : 15-38. doi: 10.3934/dcds.2007.18.15
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