Date of Award

5-2019

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Tariel Kiguradze

Second Advisor

Mark Archambault

Third Advisor

Kanishka Perera

Fourth Advisor

Gnana Bhaskar Tenali

Abstract

Two–point boundary value problems in a multidimensional box for higher order nonlinear hyperbolic equations are considered. The concepts of a strongly isolated solution, and locally and globally strong well–posedness of a nonlinear boundary value problem are introduced. For general two–point boundary value problems and periodic problems there are established: (i) Necessary and sufficient conditions of locally and globally strong well–posedness; (ii) Unimprovable Sufficient conditions of solvability. For the Dirichlet and Periodic type problems for equations of even order there are established: (i) Effective sufficient conditions of solvability and locally strong well–posedness; (ii) Unimprovable sufficient conditions of solvability for the case, where the righthand side of equation has arbitrary growth order with respect to certain phase variables; (iii) sufficient conditions of solvability and locally strong well–posedness for the case, where the righthand side of equation is H¨older continuous with respect to certain principal phase variables. For initial–boundary value problems there are established: (i) Necessary and sufficient conditions of locally and globally strong well–posedness; (ii) Unimprovable sufficient conditions of local and global solvability.

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