Boundary Value Problems in a Multidimensional Box for Higher Order Linear and Quasi-Linear Hyperbolic Equations

Abstract

Boundary value problems in a multidimensional box for higher order linear hyperbolic equations are considered. The concept of associated problems are introduced. For general boundary value problems there are established: (i) Necessary and sufficient conditions for a linear problem to have the Fredholm property in two–dimensional case; (ii) Necessary and sufficient conditions of well–posedness in two–dimensional case; (iii) Unimprovable sufficient conditions for a linear problem to have the Fredholm property; (iv) Unimprovable sufficient conditions of well–posedness and α–well–posedness; (v) Effective sufficient conditions of unqie solvability of two–point, periodic and Dirichlet type problems. (iv) Unimprovable conditions of unique solvability of two dimensional ill–posed periodic problems. For the Dirichlet type problem in a two–dimensional smooth convex domain: (i) Sufficient conditions for a linear problem to have the Fredholm property; (ii) sufficient conditions of unique solvability. For quasi–linear boundary value problems there are established: (i) Optimal sufficient conditions of solvability and unique solvability; (ii) Effective sufficient conditions of solvability of periodic and Dirichlet type problems in case, where the righthand side of the equation has arbitrary growth order with respect to some phase variables.