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Home » Issues » Archive of Issues » 2022 - 1 issue » pp. 130-149

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Russia Nizhni Novgorod
Year
2022
Volume
32
Issue
1
Pages
130-149
<<
Section Mathematics
Title On totally global solvability of evolutionary equation with monotone nonlinear operator
Author(-s) Chernov A.V.ab
Affiliations Nizhni Novgorod State Technical Universitya, Nizhni Novgorod State Universityb
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 $$x^\prime+Ax=f[u](x),\quad x(0)=a\in H;\quad x\in W=\{x\in X\colon x^\prime\in X^*\},$$ 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
DOI 10.35634/vm220109
Received 7 September 2021
Language Russian
Citation Chernov A.V. On totally global solvability of evolutionary equation with monotone nonlinear operator, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 1, pp. 130-149.
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