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

We consider the well-posedness of the abstract Cauchy problem for the doubly nonlinear evolution inclusion equation of second order given by \(\begin{aligned} \left\{ \begin{array}{ll} u''(t)+\partial \Psi (u'(t))+B(t,u(t))\ni f(t), &{} t\in (0,T),\, T>0,\\ u(0)=u_0, &{} u'(0)=v_0 \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}^{\prime \prime }(t)+\partial \mathrm{\Psi }(u^{\prime }(t))+B(t,u(t))\ni f(t),\hfill & t\in (0,T),T>0,\hfill \\ u(0)=u_0,\hfill & u^{\prime }(0)=v_0\hfill \end{array}\right)\end{array}$$ where the function \( u \) $u$ takes values in a real separable Hilbert space, denoted by \(\mathscr {H}\) $\mathcal{H}$ . Here, \( u_0 \) $u_0$ lies in \(\mathscr {H}\) $\mathcal{H}$ , \( v_0 \) $v_0$ is in the intersection \(\overline{\textrm{dom}(\partial \Psi )}\cap \textrm{dom}(\Psi )\) $\overline{\text{dom}(\partial \mathrm{\Psi })}\cap \text{dom}(\mathrm{\Psi })$ , and \( f \) $f$ belongs to \( {\mathrm L}^2(0,T;\mathscr {H}) \) ${\mathrm{L}}^2(0,T;\mathcal{H})$ . The functional \(\Psi : \mathscr {H}\rightarrow (-\infty ,+\infty ] \) $\mathrm{\Psi }:\mathcal{H}\to (-\infty ,+\infty ]$ is assumed to be proper, lower semicontinuous, and convex. Additionally, the nonlinear operator \( B:[0,T]\times \mathscr {H}\rightarrow \mathscr {H} \) $B:[0,T]\times \mathcal{H}\to \mathcal{H}$ is assumed to satisfy either a global or a local Lipschitz condition. In the case where \( B \) $B$ satisfies a global Lipschitz condition, we can establish the existence and uniqueness of strong solutions \( u \) $u$ belonging to \( {\mathrm H}^2(0,T^*;\mathscr {H}) \) ${\mathrm{H}}^2(0,T^{\ast };\mathcal{H})$ . Furthermore, these solutions continuously depend on the data. We derive these results using the theory of nonlinear semigroups combined with the Banach fixed-point theorem. On the other hand, when \( B \) $B$ satisfies a local Lipschitz condition, we can guarantee the existence of strong local solutions.