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

Abstract

This paper addresses a doubly nonlinear parabolic inclusion of the form <p align="center"> $\mathcal A (u_t)+\mathcal B (u)$ ∋ f. <p align="left" class="times"> 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 <i> unbounded</i> 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.