Well-posedness of a fully nonlinear evolution inclusion of second order
Aras Bacho
TL;DR
The paper addresses the well-posedness of the abstract second-order evolution inclusion $u''(t)+∂Ψ(u'(t))+B(t,u(t))\ni f(t)$ in a real Hilbert space $\mathscr{H}$, with $Ψ$ convex, lower semicontinuous and proper and $B$ locally Lipschitz. The authors reformulate the problem as a coupled first-order system and solve an auxiliary problem for $v$ to define a solution operator $J$, then apply a Banach fixed-point argument to the map $F(u)(t)=u_0+\int_0^t J(u)(s)ds$ to obtain a local strong solution; a global solution exists if $B$ is globally Lipschitz, together with a Gronwall-type stability estimate. The work extends well-posedness results to fully nonlinear leading parts by avoiding compactness methods, and includes an Orlicz-space based PDE example to demonstrate applicability. This provides a robust abstract framework for nonlinear second-order evolution inclusions with potential applications in nonlinear mechanics and PDEs with nonstandard growth, and suggests avenues for handling multi-valued $B$ or alternative function spaces in future work.
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 Ψ(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 $u_0\in \mathscr{H}, v_0\in \overline{D(\partial Ψ)}\cap D(Ψ), f\in L^2(0,T;\mathscr{H})$. The functional $Ψ: \mathscr{H} \rightarrow (-\infty,+\infty]$ 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.