An abstract doubly nonlinear equation with a measure as initial value
The authors are dealing with nonlinear systems of differential equations of the form \[ \begin{aligned} v'(t)+w(t)+{\mathcal C}(t)u(t)=f(t)\quad & \text{for a.a. }t\in]0,T[,\\ v(t)\in{\mathcal A}(t)u(t),\;w(t)\in{\mathcal B}(t)u(t) \quad &\text{for a.a. }t\in]0,T [,\\ v(0)=v_0,\end{aligned} \] where \(A(t)\) and \(B(t)\) are maximal monotone operators in a real Hilbert space \(V\), such that \(A(t)+B(t)\) is \(V\)-coercive. \(A(t)\) represents the subdifferential of a continuous convex function on a real Banach space \(W\), such that there exists an injection \(i:V\to W\), with \(iV\) dense in \(W\). Under technical assumptions concerning the operators involved in the system and the initial datum, two existence results are obtained and another basic result deals with regularity properties of the solution, showing function spaces to which it belongs. To carry out the proofs, the authors are establishing some auxiliary results (10 Lemmas) that require a delicate and lengthy analysis. An example is treated in detail, at the end of the paper.
scientific article; zbMATH DE number 5149565
scientific article