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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