Evolution equations governed by families of weighted operators (Q1288766)

A Lebesgue approach for a fully nonlinear nonautonomous evolution problem in an arbitrary Banach space \(X\) \[ \frac{du}{dt}+\alpha(t)A_\alpha(t)u\ni 0, \quad t\in I\subseteq[0,T], \quad u(s)=u^0, \tag{1} \] is developed. (Here, the control \(\alpha\) acts on an unbounded operator). For this purpose an abstract \(L^1\)-comparison mode (called coherence) between multivalued time dependent families of operators \((A_\alpha(s))_{s\in I}\) and \((A_\beta(t))_{t\in J}\) on compact subintervals \(I, J \subseteq[0,T]\) weighted by functions \(\alpha, \beta\in L^\infty([0,T], \mathbb{R}^+)\) is defined. The solution to the Cauchy problem (1) called mas is given as a limit of discrete implicit schemes, where approximations are in a Lebesgue sense. The main results extending Crandall's, Ligett's, Evans', and others'are (1) existence and uniqueness results: all \(\varepsilon_n\)-discrete adapted approximating families (DAF) to (1) are uniformly convergent on \(I\) towards its unique mas; (2) estimates for coherent mases are obtained; (3) \(S(t,s)\) defined as \(S(t,s)u^0=u(t)\) is proved to be an evolution operator for a strongly \(\psi\)-coherent family \((A_\alpha)_{[0,T]}\).