Doubly nonlinear evolution equations with non-monotone perturbations in reflexive Banach spaces
The paper is concerned with the Cauchy problem for a doubly nonlinear evolution equation \[ \begin{cases}\partial_V\psi^t(u'(t))+\partial_V\varphi(u(t))+B(t,u(t))\ni f(t)\,\,\,\text{in}\,\,\,V^*,\,\,\,0<t<T,\\ u(0)=u_0,\end{cases} \] where \(V\) and \(V^*\) is a reflexive Banach space and its dual space, \(V\hookrightarrow H\equiv H^*\hookrightarrow V^*\) with continuous and densily defined canonical injections, \(H\) is a Hilbert space identified with its dual space, \(\partial _V\psi^t\) and \(\partial_V\varphi:V\to 2^{V^*}\) are the subdifferential operators of proper, lower semicontinuous and convex functions \(\psi^t\) and \(\varphi\), respectively, from \(V\) into \((-\infty,+\infty]\), \(u'(t)=du(t)/dt\), \(f:(0,T)\to V^*\) and \(u_0\in V\) are given, and \(B:(0,T)\times V\to V^*\) is a possibly multi-valued mapping such that \(B(t,\cdot)\) might be non-monotone in \(V\times V^*\) for each fixed \(t\). By using a couple of approximations for the equation and a fixed-point argument with a multi-valued mapping, the author gives sufficient conditions for the local (in time) existence of strong solutions to the above problem and for the global existence. The presented framework can cover evolution equations whose solutions might blow up in finite time and whose unperturbed equations (i.e. \(B\equiv 0\)) might not be uniquely solved in a doubly nonlinear setting. Some applications of the resulting theorems to doubly nonlinear parabolic equations are also presented.
scientific article; zbMATH DE number 5997662
scientific article