An implicit nonlinear time dependent equation has a solution
It will be shown that the differential equation \(v'(t)+w(t)+{\mathcal C}(t)u(t)=f(t)\) for a.e. \(t\in]0,T[\), \(v(t)\in{\mathcal A}(t)u(t)\), \(w(t)\in{\mathcal B}(t)u(t)\), for a.e. \(t\in]0,T[\), and \(v(0)=v_ 0\), has a solution \((u,v,w)\). The operators \({\mathcal A}(t)\) and \({\mathcal B}(t)\) are maximal monotone from a real Hilbert space \(V\) to its dual such that \({\mathcal A}(t)+{\mathcal B}(t)\) are \(V\)-coercive. A linear compact injection \(i\) embeds \(V\) to a real Banach space \(W\) and each \({\mathcal A}(t)\) is the strongly-monotone subdifferential of a continuous convex function \(\phi(t,.)\) on \(W\). The function \(f\) is square integrable. The functions \({\mathcal C}(t): V\to W^*\) are Lipschitzian as \(V^*\)-valued functions. As example an implicit nonlinear parabolic PDE with time dependent coefficients is considered. The result implies the existence of its weak solution.
scientific article; zbMATH DE number 57295
scientific article