Well-posedness of a fully nonlinear evolution inclusion of second order

TL;DR

This paper proves the existence, uniqueness, and continuous dependence of solutions for a second-order doubly nonlinear evolution inclusion in Hilbert spaces, extending the theory of nonlinear semigroups.

Contribution

It establishes well-posedness results for a fully nonlinear second-order evolution inclusion with a convex potential and a Lipschitz nonlinear operator.

Findings

Existence of strong solutions under global Lipschitz conditions.

Uniqueness and continuous dependence of solutions.

Local solutions when the nonlinear operator is only locally Lipschitz.

Abstract

The well-posedness of the abstract \textsc{Cauchy} problem for the doubly nonlinear evolution inclusion equation of second order \begin{align*} \begin{cases} u''(t)+\partial \Psi(u'(t))+B(t,u(t))\ni f(t), &\quad t\in (0,T),\, T>0,\\ u(0)=u_0, \quad u'(0)=v_0 \end{cases} \end{align*} in a real separable \textsc{Hilbert} space $\mathscr{H}$, where $u_0\in \mathscr{H}, v_0\in \overline{D(\partial \Psi)}\cap D(\Psi), f\in L^2(0,T;\mathscr{H})$. The functional $\Psi: \mathscr{H} \rightarrow (-\infty,+\infty]$ is supposed to be proper, lower semicontinuous, and convex and the nonlinear operator $B:[0,T]\times \mathscr{H}\rightarrow \mathscr{H}$ is supposed to satisfy a (local) \textsc{Lipschitz} condition. Existence and uniqueness of strong solutions $u\in H^2(0,T^*;\mathscr{H})$ as well as the continuous dependence of solutions from the data re shown by employing the theory of nonlinear semigroups and the Banach fixed-point theorem. If $B$ satisfies a local Lipschitz condition, then the existence of strong local solutions are obtained.