On existence and uniqueness of solution of hyperbolic differential inclusion with discontinuous nonlinearity
The existence and uniqueness of solutions are presented to the differential inclusion \[ u''(t)+ Bu(t)+\varphi \bigl(u'(t) \bigr)\ni f(t), \quad \text{a.e. }t \in[0,T], \] \[ u(x,t)=0,\quad \text{a.e. } (x,t) \in\partial \Omega\times [0,T], \quad u(0)=u_0,\;u'(0)=u_1, \] in the triple \(V:=H^1_0(\Omega)\subset L^2(\Omega)\subset H^{-1}(\Omega)=:V'\) with \(\Omega\) a bounded open set in \(\mathbb{R}^n\), \(u_0\in V\), \(u_1\in L^2(\Omega)\). Here, \(B\) is a linear operator in \(L^2(\Omega)\) with domain \(V\), \(\varphi (\zeta)=[\underline b(\xi), \overline b(\xi)]\) with \(\underline b,\overline b\) functions generated by a given function \(b\in L^\infty_{\text{loc}}\) in some way, and \(\partial \varphi\) means the Clark subdifferential of \(\varphi\). The asymptotic behaviour of the solution is also given.
scientific article; zbMATH DE number 1423859