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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.

##### MSC:
 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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