Some classes of nonstationary equations with growing lower terms
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.
superconductor-dielectric system with tunnel effect
scientific article; zbMATH DE number 1285994