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Evolution equations governed by families of weighted operators. (English) Zbl 0926.34051
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 \((A_\alpha)_{[0,T]}\).

MSC:
34G20 Nonlinear differential equations in abstract spaces
35G10 Initial value problems for linear higher-order PDEs
35K25 Higher-order parabolic equations
47J05 Equations involving nonlinear operators (general)
47N40 Applications of operator theory in numerical analysis
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