Geodesic
Mixed problem for a higher-order nonlinear pseudoparabolic equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical Analysis, Tome 156 (2018), pp. 73-83.

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We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters.
Mots-clés : pseudoparabolic equation
Keywords: generalized derivative, method of successive approximations, parameter, unique solvability. 
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T. K. Yuldashev; K. H. Shabadikov. Mixed problem for a higher-order nonlinear pseudoparabolic equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical Analysis, Tome 156 (2018), pp. 73-83. http://geodesic.mathdoc.fr/item/INTO_2018_156_a5/

[1] Aleksandrov V. M., Kovalenko E. V., Zadachi mekhaniki sploshnykh sred so smeshannymi granichnymi usloviyami, Nauka, M., 1986

[2] Algazin S. D., Kiiko I. A., Flatter plastin i obolochek, Nauka, M., 2006

[3] Zamyshlyaeva A. A., “Matematicheskie modeli sobolevskogo tipa vysokogo poryadka”, Vestn. Yuzhno-Uralsk. gos. un-ta. Ser. Mat. model. programmir., 7:2 (2014), 5–28 | Zbl

[4] Ilin V. A., “O razreshimosti smeshannykh zadach dlya giperbolicheskogo i parabolicheskogo uravnenii”, Usp. mat. nauk, 15:2 (92) (1960), 97–154 | Zbl

[5] Ilin V. A., Moiseev E. I., “Optimizatsiya za proizvolnyi dostatochno bolshoi promezhutok vremeni granichnogo upravleniya kolebaniyami struny uprugoi siloi”, Differ. uravn., 42:12 (2006), 1699–1711 | MR | Zbl

[6] Karimov Sh. T., “Ob odnom metode resheniya zadachi Koshi dlya odnomernogo polivolnovogo uravneniya s singulyarnym operatorom Besselya”, Izv. vuzov. Mat., 2017, no. 8, 27–41 | Zbl

[7] Koshanov B. D., Soldatov A. P., “Kraevaya zadacha s normalnymi proizvodnymi dlya ellipticheskogo uravneniya vysokogo poryadka na ploskosti”, Differ. uravn., 52:12 (2016), 1666–1681 | DOI | MR | Zbl

[8] Pokhozhaev S. I., “O razreshimosti kvazilineinykh ellipticheskikh uravnenii proizvolnogo poryadka”, Mat. sb., 117:2 (1982), 251–265 | MR | Zbl

[9] Skrypnik I. V., Nelineinye ellipticheskie uravneniya vysshego poryadka, Naukova dumka, Kiev, 1973

[10] Todorov T. G., “O nepreryvnosti ogranichennykh obobschennykh reshenii kvazilineinykh ellipticheskikh uravnenii vysokogo poryadka”, Vestn. Leningradsk. gos. un-ta, 19 (1975), 56–63 | Zbl

[11] Yuldashev T. K., “Smeshannaya zadacha dlya nelineinogo integrodifferentsialnogo uravneniya s parabolicheskim operatorom vysokoi stepeni”, Zh. vychisl. mat. mat. fiz., 52:1 (2012), 112–123 | MR | Zbl

[12] Yuldashev T. K., “Obratnaya zadacha dlya nelineinogo integrodifferentsialnogo uravneniya s giperbolicheskim operatorom vysokoi stepeni”, Vestn. Yuzhno-Uralsk. gos. un-ta. Ser. Mat. Mekh. Fiz., 5:1 (2013), 69–75 | Zbl

[13] Yuldasheva A. V., “Ob odnoi zadache dlya kvazilineinogo uravneniya chetnogo poryadka”, Differentsialnye uravneniya. Matematicheskaya fizika, Itogi nauki i tekhn. Sovr. mat. prilozh. Temat. obzory, 140, VINITI RAN, M., 2017, 43–49 | MR

[14] Karimov Sh. T., “Multidimensional generalized Erdélyi–Kober operator and its application to solving Cauchy problems for differential equations with singular coefficients”, Fract. Calc. Appl. Anal., 18:4 (2015), 845–861 | MR | Zbl