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On Coercive Solvability of Parabolic Equations with Variable Operators

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Abstract

In a Banach space E, the Cauchy problem

$$ \upsilon^{\prime }(t)+A(t)\upsilon (t)=f(t)\kern1em \left(0\le t\le 1\right),\kern1em \upsilon (0)={\upsilon}_0, $$

is considered for a differential equation with linear strongly positive operator A(t) such that its domain D = D(A(t)) does not depend on t and is everywhere dense in E and A(t) generates an analytic semigroup exp{−sA(t)}(s ≥ 0). Under natural assumptions on A(t), we prove the coercive solvability of the Cauchy problem in the Banach space \( {C}_0^{\beta, \upgamma} \) (E). We prove a stronger estimate for the solution compared with estimates known earlier, using weaker restrictions on f(t) and v0.

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Authors and Affiliations

  1. RUDN University, Moscow, Russia

    A. R. Hanalyev

Authors
  1. A. R. Hanalyev
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Correspondence to A. R. Hanalyev.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 61, Proceedings of the Crimean Autumn Mathematical School-Symposium, 2016.

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Hanalyev, A.R. On Coercive Solvability of Parabolic Equations with Variable Operators. J Math Sci 239, 706–724 (2019). https://doi.org/10.1007/s10958-019-04321-x

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  • DOI: https://doi.org/10.1007/s10958-019-04321-x

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