Existence for second order differential inclusions on \({\mathbb R}_+\) governed by monotone operators (Q2923158)

The author studies a second order differential inclusion in a Hilbert space \(H\) of the form NEWLINE\[NEWLINEp(t)u''(t)+q(t)u'(t) \in Au(t)+f(t)\quad(t\in\, ]0,\infty[\, \text{ a.e.}),\, u(0) =x.NEWLINE\]NEWLINE Here \(A\) is a maximal monotone set-valued operator on \(H\) satisfying \(0\in R(A)\), \(p,q\in L_{\infty}([0,\infty[;\mathbb{R})\) with \(p\) strictly positive and \(q\) either strictly positive or strictly negative, \(x\in\overline{D(A)}\) and \(f\) is a given source term in a weighted \(L_{2}\)-space. The author proves the existence and uniqueness of a strong solution \(u\) and thereby generalizes a result by himself [the author, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 83, 69--81 (2013; Zbl 1267.47110)], where the existence and uniqueness was proved for \(x\in D(A).\) Thus, the presented result in particular implies that the differential inclusion shows a smoothing effect in the sense that a solution with initial condition in \(\overline{D(A)}\) becomes an element of \(D(A)\) for almost every \(t>0\).