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Two remarks on the generalised Korteweg de-Vries equation
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 |
$\mathcal A (u_t)+\mathcal B (u)$ ∋ 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.
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