Study of classical solution of a one-dimensional mixed problem for one class of fifth-order semilinear equations of the Korteweg–de Vries–Burgers type

As is well known, many problems of mathematical physics are reduced to one- and multi-dimensional initial and initial–boundary value problems for, generally speaking, strongly nonlinear pseudoparabolic equations. The existence (local and global) and uniqueness of a classical solution to a one-dimensional mixed problem with homogeneous Riquier-type boundary conditions are analyzed for a class of fifth-order semilinear pseudoparabolic equations of the Korteweg–de Vries–Burgers type. For the classical solution of the mixed problem, a uniqueness theorem is proved using the Gronwall–Bellman inequality, a local existence theorem is proved by combining the generalized contraction mapping principle with the Schauder fixed point principle, and a global existence theorem is proved by applying the method of a priori estimates.