Preservation of global solvability and estimation of solutions of some controlled nonlinear partial differential equations of the second order

A. V. Chernov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 34 (2024) no. 4, pp. 541-562

Résumé

Let $U$ be the set of admissible controls, $T>0$, and let $W[0;\tau]$, $\tau\in(0;T]$, be a scale of Banach spaces such that the set of restrictions of functions from $W=W[0;T]$ to $[0;\tau]$ coincides with $W[0;\tau]$; let $F[.;u]\colon W\to W$ be a controlled Volterra operator, $u\in U$. Earlier, for the operator equation $x=F[x;u]$, $x\in W$, the author introduced a comparison system in the form of a functional integral equation in the space $\mathbf{C}[0;T]$. It was established that to preserve (under small perturbations of the right-hand side) the global solvability of the operator equation, it is sufficient to preserve the global solvability of the specified comparison system, and the corresponding sufficient conditions were established. In this paper, further examples of application of this theory are considered: nonlinear wave equation, strongly nonlinear wave equation, nonlinear heat equation, strongly nonlinear parabolic equation.

Détail

Keywords: second kind evolutionary Volterra equation of general form, functional integral equation, comparison system, preservation of global solvability, uniqueness of solution, nonlinear wave equation, nonlinear parabolic equation