Some classes of nonstationary equations with growing lower terms
type: Item
instance of: scholarly article
title: Some classes of nonstationary equations with growing lower terms
zbMATH Open document ID: 0917.35005
DOI: 10.1007/BF02673057
author: A. I. Kozhanov
published in: Siberian Mathematical Journal
publication date: 1999-05-11T00:00:00Z
review text: The article under review is devoted to studying solvability and properties of solutions to the initial-boundary value problem for equations \[ u_{tt} - \Delta u - \lambda\Delta u_t + g(u,u_t) = f(x,t), \] where \(\Delta\) is the Laplace operator in the variables \(x=(x_1,\dots, x_n)\), \(\lambda = \text{const}\geq 0\) (in the case \(\lambda >0\) such equations are sometimes called pseudohyperbolic), and the nonlinear function \(g(\xi,\eta)\) models the function \(g(\xi,\eta) = a(\xi)| \eta| ^{p-2}\eta\), \(a(\xi)\geq 0\), \(p>1\). In the model case, these equations arise in the description of the motion of electrons in the superconductor-dielectric system with tunnel effect (the Josephenson contact or the Josephenson power transmission line). For the model equation, the case \(a(\xi)\equiv \text{const}>0\) was studied by J.-L. Lions in detail for \(\lambda =0\) and by the author [Math. USSR, Sb. 46, 507-525 (1983; Zbl 0564.35019)] for \(\lambda > 0\). If \(a\not\equiv\text{const}\) then the compactness arguments yield existence of a global solution under the smallness assumptions for the right-hand side and initial data. Regarding the general case of a nonlinear function \(g(\xi,\eta)\) (which covers the physics case), in the article under review, the author proposes some methods based on the compound structure of the equation which allow us to prove global solvability of initial-boundary value problem without smallness assumptions. Alongside the solvability results, the properties of solutions for the problem under consideration are also exposed.
Mathematics Subject Classification ID: 35A05
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35M20
Mathematics Subject Classification ID: 35R60
zbMATH DE Number: 1285994
zbMATH Keywords: nonlinear equation of composite type
zbMATH Keywords: pseudohyperbolic equation
zbMATH Keywords: initial-boundary value problem
zbMATH Keywords: global existence
zbMATH Keywords: superconductor-dielectric system with tunnel effect
zbMATH Keywords: Josephenson power transmission line
reviewed by: Vladimir N. Grebenev
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf02673057
OpenAlex ID: W2076066280
description: scientific article; zbMATH DE number 1285994