Inverse Problem for an Equation with a Nonstandard Growth Condition
S. N. Antontsev1, S. E. Aitzhanov2
1Lavrent'ev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sceinces, Novosibirsk, 630090, Russia
2Al-Farabi Kazakh National University, Almaty, 050038, Kazakhstan
Keywords: обратная задача, интегральное условие переопределения, параболические уравнения с нестандартным условием роста, разрешимость, разрушение решения, асимптотическое поведение решения, inverse problem, integral overdetermination condition, parabolic equations with a nonstandard growth condition, solvability, blow-up of the solution, asymptotic solution behavior
Abstract
This paper describes an inverse problem for determining the right side of a parabolic equation with a nonstandard growth condition and integral overdetermination condition. The Galerkin method is used to prove the existence of two solutions of the inverse problem and their uniqueness, one of them being local and the other one being global in time. Sufficient blow-up conditions for the local condition for a finite time in a limited region with a homogeneous Dirichlet condition on its boundary. The blow-up of the solution is proven using the Kaplan method. The asymptotic behavior of the inverse problem solutions for large time values is investigated. Sufficient conditions for vanishing of the solution for a finite time are obtained. Boundary conditions ensuring the corresponding behavior of the solutions are considered.
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