Boundary-value problems for one class of composite equations with the wave operator in the senior part

A. I. Kozhanov ; T. P. Plekhanova

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Mathematical Modeling, Tome 188 (2020), pp. 76-83

Résumé

The work is devoted to the solvability of local and nonlocal boundary-value problems for composite (Sobolev-type) equations $ D^{2p+1}_t\left(D^2_t-\Delta u \right) + Bu = f(x,t), $ where $D^k_t={\partial^k}/{\partial t^k}$, $\Delta$ is the Laplace operator acting on spatial variables, $B$ is a second-order differential operator that also acts on spatial variables, and $p$ is a nonnegative integer. For these equations, the existence and uniqueness of regular solutions (possessing all generalized derivatives in the Sobolev sense that are involved in the equation) to initial-boundary-value problems and the boundary-value problems nonlocal in the time variable. Some generalizations and refinements of the results obtained are also described.

Détail

Mots-clés : composite equation, existence
Keywords: wave operator, initial-boundary-value problem, nonlocal boundary-value problem, regular solution, uniqueness.