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Relaxed solutions of a Sobolev-type linear equation and semigroups of operators. (English. Russian original) Zbl 1090.47027
Izv. Math. 67, No. 4, 797-813 (2003); translation from Izv. Ross. Akad. Nauk Ser. Mat. 67, No. 4, 171-188 (2003).
The author considers the Cauchy problem of Sobolev type \(L\frac{d}{dt} u(t) = M u(t)\) under the initial condition \(u(0) = u_0\) on a bounded closed interval \([0, T]\). He shows that if the pair \((M, L)\) satisfies the \((p, \phi (\eta ))\)-condition introduced in this paper, then there is an infinitely differentiable semigroup for the Cauchy problem. Under the strong \((p, \phi (\eta ))\)-condition, the set of one-valued solubility of the weakened Cauchy problem for this equation is found. This generalizes the previously known results in this direction. Some examples satisfying the \((p, \phi (\eta ))\)-condition are given.

47D06 One-parameter semigroups and linear evolution equations
34G10 Linear differential equations in abstract spaces
35K90 Abstract parabolic equations
Zbl 0944.47025
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