Time discretization of nonlinear Cauchy problems applying to mixed hyperbolic-parabolic equations
The authors consider the Cauchy evolution problem \[ Lu_{tt}+Bu_t+Au\ni f\quad\text{in }(0,T),\quad u(0)=u_0,\quad u_t(0)=v_0. \] Here \(L:H\to H\) and \(A:V\to V'\) are two linear, bounded selfadjoint operators, where \(V\), \(H\) are Hilbert spaces such that \(V\subset H\) is continuously imbedded and dense. \(B\) is a maximal monotone operator from \(V\) to \(V'\). \(L\) may be degenerate but the sum \(L+B\) is assumed to be coercive in \(H\). The condition on the map \(\alpha I+A\) is to be strongly monotone from \(V\) to \(V'\) for all \(\alpha>0\), where \(I\) denotes the identity in \(H\). The case where \(L=I\) has previously been studied e.g. by \textit{J.-L. Lions} and \textit{W. A. Strauss} [Bull. Soc. Math. Fr. 93, 43-96 (1965; Zbl 0132.10501)]. The authors prove the existence and uniqueness of a variational solution of the given problem. The existence proof is obtained by discretizing the problems with respect to time with the backward Euler method. The Euler approximations are shown to converge with order \(O(\tau^{1/2})\), where \(\tau\) is the time increment.
scientific article; zbMATH DE number 894080