Boundary value problems for third-order pseudoelliptic equations with degeneration

A. I. Kozhanov

Matematičeskie zametki SVFU, Tome 28 (2021) no. 1, pp. 27-36

Résumé

We study the solvability of boundary value problems in cylindrical domains $Q=\Omega\times(0,T)$, $\Omega\subset\mathbb{R}^n$, $0<T<+\infty$, for differential equations $$ h(t)\frac{\partial^{2p+1}u}{\partial t^{2p+1}}+(-1)^{p+1}\Delta u+c(x,t)u=f(x,t), $$ where $p$ is a non-negative integer, $h(t)$ is continuous on the segment $[0, T]$ a function such that $\varphi(t)>0$ for $t\in(0,T)$, $\varphi(0)\ge0$, $\varphi(T)\ge0$, and $\Delta$ is the Laplace operator in spatial variables $x_1,\dots, x_n$. The main feature of the problems under study is that, despite the degeneration, the boundary manifolds are not exempt to the bearing boundary conditions. We proved the existence and uniqueness theorems of the regular solutions, those having all Sobolev generalized derivatives included in the equation. Moreover, we describe some possible enhancements and generalizations of the obtained results.

Détail

Mots-clés : quasi-parabolic equations, existence
Keywords: degeneration, boundary value problem, regular solution, uniqueness.