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Abstract evolution differential equations with discontinuous operator coefficients. (English. Russian original) Zbl 0868.47012
Differ. Equations 31, No. 7, 1067-1076 (1995); translation from Differ. Uravn. 31, No. 7, 1132-1141 (1995).
The paper is devoted to studying the Cauchy problem for the abstract evolution differential equation $$du/dt+ A(t)u=f$$ in a Hilbert space $$H$$, where $$A(t)$$ is a linear unbounded operator in $$H$$ with coefficients discontinuous on a set of zero measure, $$u$$ and $$f$$ are $$H$$-valued functions of the variable $$t\in]0,T[$$. The main theorems proved give a sufficient condition for
1) the existence and uniqueness of strong solutions of the above problem, and
2) their smoothness in scales of various spaces.
In conclusion, the author applies the received results to the specific mixed problems earlier unstudied for variable-order parabolic equations with time-discontinuous coefficients both in equations and in normal boundary conditions.

##### MSC:
 47A50 Equations and inequalities involving linear operators, with vector unknowns 35K30 Initial value problems for higher-order parabolic equations 47D06 One-parameter semigroups and linear evolution equations 34G10 Linear differential equations in abstract spaces