Investigation of the classical solution of a 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 {\footnotesize
$$ U_{tt}(t, x)-U_{xx}(t, x)-\alpha U_{ttxx}(t, x)=F(t, x, U(t, x), U_x(t, x), U_{xx}(t, x), U_t(t, x), U_{tx}(t, x), U_{txx}(t, x)), $$
} where $\alpha>0$ is a fixed number, $0\leq t\leq T$, $0\leq x\leq\pi$, $0<T<+\infty$, $F$ 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.