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JOURNALS // Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki // Archive

Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2022 Volume 32, Issue 1, Pages 130–149 (Mi vuu803)

This article is cited in 1 paper

MATHEMATICS

On totally global solvability of evolutionary equation with monotone nonlinear operator

A. V. Chernovab

a Nizhny Novgorod State Technical University, ul. Minina, 24, Nizhny Novgorod, 603950, Russia
b National Research Lobachevsky State University of Nizhni Novgorod, pr. Gagarina, 23, Nizhny Novgorod, 603950, Russia

Abstract: Let $V$ be a separable reflexive Banach space being embedded continuously in a Hilbert space $H$ and dense in it; $X=L_p(0,T;V)\cap L_{p_0}(0,T;H)$; $U$ be a given set of controls; $A\colon X\to X^*$ be a given Volterra operator which is radially continuous, monotone and coercive (and, generally speaking, nonlinear). For the Cauchy problem associated with controlled evolutionary equation as follows
$$x^\prime+Ax=f[u](x), x(0)=a\in H; x\in W=\{x\in X\colon x^\prime\in X^*\},$$
where $u\in U$ is a control, $f[u]\colon \mathbf{C}(0,T;H)\to X^*$ is Volterra operator ($W\subset\mathbf{C}(0,T;H)$), we prove totally (with respect to a set of admissible controls) global solvability subject to global solvability of some functional integral inequality in the space $\mathbb{R}$. In many particular cases the above inequality may be realized as the Cauchy problem associated with an ordinary differential equation. In fact, a similar result proved by the author earlier for the case of linear operator $A$ and identity $V=H=V^*$ is developed. Separately, we consider the cases of compact embedding of spaces, strengthening of the monotonicity condition and coincidence of the triplet of spaces $V=H=H^*$. As to the last two cases, we prove also the uniqueness of the solution. In the first case we use Schauder theorem and in the last two cases we apply the technique of continuation of solution along with the time axis (i. e., continuation along with a Volterra chain). Finally, we give some examples of an operator $A$ satisfying our conditions.

Keywords: strongly nonlinear evolutionary equation in a Banach space, monotone nonlinear operator, totally global solvability.

UDC: 517.957, 517.988, 517.977.56

MSC: 47J05, 47J35, 47N10

Received: 07.09.2021
Accepted: 05.01.2022

DOI: 10.35634/vm220109



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© Steklov Math. Inst. of RAS, 2024