Geodesic
Investigation of the classical solution of a one-dimensional mixed problem for a class of semilinear long-wave equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 9, pp. 1569-1586 
F. M. Namazov; K. I. Khudaverdiev. Investigation of the classical solution of a one-dimensional mixed problem for a class of semilinear long-wave equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 9, pp. 1569-1586. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_9_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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, $F$ 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.

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