Study of the classical solution to the one-dimensional mixed problem for a class of semilinear long-wave equations

Many problems in mathematical physics are reduced to one- or multidimensional initial and initial-boundary value problems for, generally speaking, strongly nonlinear Sobolev-type equations. In this work, local and global classical solvability is studied for the one-dimensional mixed problem with homogeneous Riquier-type boundary conditions for a class of semilinear long-wave equations \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ U_{tt} \left( {t,x} \right) - U_{xx} \left( {t,x} \right) - \alpha U_{ttxx} \left( {t,x} \right) = F\left( {t,x,U\left( {t,x} \right),U_x \left( {t,x} \right),U_{xx} \left( {t,x} \right),U_t \left( {t,x} \right),U_{tx} \left( {t,x} \right),U_{txx} \left( {t,x} \right)} \right) $$\end{document}, where α > 0 is a fixed number, 0 ≤ t ≤ T, 0 ≤ x ≤ π, 0 < T < +∞, F is a given function, and U(t, x) is the sought function. A uniqueness theorem for the mixed problem is proved using the Gronwall-Bellman inequality. A local existence result is proved by applying the generalized contraction mapping principle combined with the Schauder fixed point theorem. The method of a priori estimates is used to prove the global existence of a classical solution to the mixed problem.