Evolution equations with monotone operator and functional nonlinearity at the time derivative (Q2710724)

The author investigates doubly nonlinear evolution equations of the form \(Au+(\partial / \partial t)Bu = f(t)\) with \(u(0)=u_0\), where \(A\) and \(B\) are monotone operators. More precisely, \(A\) is induced by a differential expression containing higher-order partial derivatives and \(B\) is induced by a real monotone function defined on the real axis. An existence theorem is proved by using both monotonicity and compactness arguments. An application for nonlinear parabolic equations of the form NEWLINE\[NEWLINE \sum_{|\alpha |\leq m}D^{\alpha}A_{\alpha}(t,x,u, \cdots , D^mu) +\beta (u)\frac{\partial u}{\partial t} = f, \;\;u(0,x)=u_0(x) NEWLINE\]NEWLINE is given.