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 Cet article a éte moissonné depuis la source Math-Net.Ru

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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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     title = {Investigation of the classical solution of a one-dimensional mixed problem for a class of semilinear long-wave equations},
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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/

[1] Radochova V., “Einige Eigenschaften der losung der Wellengleichung mit der Korrektion von Rayleigh”, Arch. Mat., 8:3 (1972), 125–129 | MR

[2] Radochova V., “Die losung der partiellen Differentialgleichung $U_{xxtt} = A(t, x)U_{xx} + B(t, x)U_{tt}$ mit gewissen Nebenbedingungen”, Cas. Pestov. Math., 98:4 (1973), 389–399 | MR

[3] Seyler C. E., Fenstermacher D. L., “A symmetric-regularized-long-wave equation”, Phys. Fluids, 27:1 (1984), 4–7 | DOI | Zbl

[4] Clarkson P. A., Le Vegue R. J., Saxton K., “Solitary-wave interactions in elastic rods”, Stud. Appl. Math., 75:2 (1986), 95–121 | MR | Zbl

[5] Gabov S. A., Orazov B. B., “Ob uravnenii $\frac{\partial^2}{\partial t^2}[U_{xx}-U]+U_{xx}=0$ i nekotorykh svyazannykh s nim zadachakh”, Zh. vychisl. matem. i matem. fiz., 26:1 (1986), 92–102 | MR | Zbl

[6] Muto V., Halding I., Christiansen P. L., Scott A. C., “A solution model for microwave absorption in DNA”, IMACS Ann. Comput. and Appl. Math., 1–4, no. 1, 1989, 119–122 | MR

[7] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Pletner Yu. D., Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, M., 2007

[8] Alshin A. B., Pletner Yu. D., Sveshnikov A. G., “Odnoznachnaya razreshimost zadachi Dirikhle dlya uravneniya ionno-zvukovykh voln v plazme”, Dokl. RAN, 361:6 (1998), 749–751 | MR

[9] Khudaverdiev K. I., Namazov F. M., “Issledovanie resheniya pochti vsyudu odnomernoi smeshannoi zadachi dlya odnogo klassa polulineinykh simmetrichno regulyarizovannykh uravnenii dlinnykh voln”, Tr. Azerbaidzhanskogo matem. ob-va, 2, Baku, 1996, 198–222 | MR

[10] Khudaverdiev K. I., “Issledovanie klassicheskogo resheniya mnogomernoi smeshannoi zadachi dlya odnogo klassa kvazilineinykh giperbolicheskikh uravnenii vtorogo poryadka”, Uch. zap. Az. gosun-ta. Ser. fiz.-matem. nauk, 1972, no. 1, 3–27 | MR

[11] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, Gostekhteorizdat, M., 1956 | MR