Conditions for the global solvability of the Cauchy problem for a semilinear equation of pseudoparabolic type

M. O. Korpusov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 43 (2003) no. 8, pp. 1210-1222

Bibliographie

[1] Bonch-Bruevich V. L., Kalashnikov S. G., Fizika poluprovodnikov, Nauka, M., 1990

[2] Sviridyuk G. A., “K obschei teorii polugrupp operatorov”, Uspekhi matem. nauk, 49:4 (1994), 47–74 | MR | Zbl

[3] Sviridyuk G. A., Fedorov V. E., “Analiticheskie polugruppy s yadrami i lineinye uravneniya tipa Soboleva”, Sibirskii matem. zhurn., 36:5 (1995), 1130–1145 | MR | Zbl

[4] Sviridyuk G. A., Semenova I. N., “O razreshimosti neodnorodnoi zadachi dlya obobschennogo filtratsionnogo uravneniya Bussineska”, Differents. ur-niya, 24:9 (1988), 1607–1613 | MR

[5] V. E. Fedorov (red.), Uravneniya Sobolevskogo tipa, Sbornik nauchnykh rabot, Chelyab. gos. un-t, Chelyabinsk, 2002 | MR

[6] Egorov I. E., Pyatkov S. G., Popov S. V., Neklassicheskie differentsialno-operatornye uravneniya, Nauka, Novosibirsk, 2000 | MR

[7] Favini A., Yagi A., Degenerate differential equations in Banach spaces, Marcel Dekker Inc., New York etc., 1999 | MR | Zbl

[8] Demidenko G. V., Uspenskii S. V., Uravneniya i sistemy, ne razreshennye otnositelno starshei proizvodnoi, Nauchn. kniga, Novosibirsk, 1998 | MR | Zbl

[9] Gaevskii Kh., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR

[10] Stefanelli U., “On a class of doubly nonlinear nonlocal evolution equations”, Different. Integral Equat., 15:8 (2002), 897–922 | MR | Zbl

[11] Showalter R. E., Monotone operators in Banach space and nonlinear partial differential equations, Math. Surveys and Monographs, 49, Amer. Math. Soc., 1997 | MR | Zbl

[12] Lyashko S. I., Obobschennoe upravlenie lineinymi sistemami, Nauk. dumka, Kiev, 1998 | Zbl

[13] Kozhanov A. I., “Parabolicheskie uravneniya s nelineinym nelokalnym istochnikom”, Sibirskii matem. zhurnal, 35:5 (1994), 1062–1073 | MR | Zbl

[14] Kozhanov A. I., “Nachalno-kraevaya zadacha dlya uravnenii tipa obobschennogo uravneniya Bussineska s nelineinym istochnikom”, Matem. zametki, 65:1 (1999), 70 | MR | Zbl

[15] Korpusov M. O., Pletner Yu. D., Sveshnikov A. G., “O kvazistatsionarnykh protsessakh v provodyaschikh sredakh bez dispersii”, Zh. vychisl. matem. i matem. fiz., 40:8 (2000), 1237–1249 | MR | Zbl

[16] Korpusov M. O., Sveshnikov A. G., “O razrushenii za konechnoe vremya resheniya nachalno-kraevoi zadachi dlya polulineinogo uravneniya sostavnogo tipa”, Zh. vychisl. matem. i matem. fiz., 40:11 (2000), 1647–1654 | MR | Zbl

[17] Korpusov M. O., “Globalnaya razreshimost i razrushenie za konechnoe vremya reshenii nelineinykh uravnenii psevdoparabolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 42:6 (2002), 849–866 | MR | Zbl

[18] Korpusov M. O., “K voprosu o razrushenii za konechnoe vremya resheniya zadachi Koshi dlya psevdoparabolicheskogo uravneniya $Au_t=F(u)$”, Differents. ur-niya, 38:12 (2002), 1–6 | MR

[19] Korpusov M. O., Sveshnikov A. G., “Energeticheskaya otsenka pri $t\to\infty$ dlya resheniya nelineinogo uravneniya psevdoparabolicheskogo tipa”, Zh. vychisl. matem. i matem. fiz., 42:8 (2002), 1200–1206 | MR | Zbl

[20] Korpusov M. O., ““Razrushenie” resheniya psevdoparabolicheskogo uravneniya s proizvodnoi po vremeni ot nelineinogo ellipticheskogo operatora”, Zh. vychisl. matem. i matem. fiz., 42:12 (2002), 1788–1795 | MR | Zbl

[21] Korpusov M. O., Sveshnikov A. G., “O suschestvovanii resheniya uravneniya Laplasa s nelineinym dinamicheskim granichnym usloviem”, Zh. vychisl. matem. i matem. fiz., 43:1 (2003), 95–110 | MR

[22] Korpusov M. O., Sveshnikov A. G., “O razreshimosti silno nelineinogo uravneniya psevdoparabolicheskogo tipa s dvoinoi nelineinostyu”, Zh. vychisl. matem. i matem. fiz., 43:7 (2003) | MR

[23] Gladkov A. L., “Edinstvennost resheniya zadachi Koshi dlya nekotorykh kvazilineinykh psevdoparabolicheskikh uravnenii”, Matem. zametki, 60:3 (1996), 356 | MR | Zbl

[24] Gabov S. A., Sveshnikov A. G., Lineinye zadachi teorii nestatsionarnykh vnutrennikh voln, Nauka, M., 1990 | MR

[25] Pletner Yu. D., “Fundamentalnye resheniya operatorov tipa Soboleva i nekotorye nachalno-kraevye zadachi”, Zh. vychisl. matem. i matem. fiz., 32:12 (1992), 1885–1899 | MR | Zbl

[26] Pyatkov S. G., “Kraevye zadachi dlya nekotorykh uravnenii i sistem, voznikayuschikh v teorii elektricheskikh tsepei”, Aktualnye vopr. sovrem. matem., 1995, no. 1, 121–133 | Zbl

[27] Guowang C., Shubin W., “Existence and non-existence of global solutions for nonlinear hyperbolic equations of higher order”, Comment. Math. Univ. Carolinae, 36:3 (1995), 475–487 | MR | Zbl

[28] Gabov S. A., Novye zadachi matematicheskoi teorii voln, Fizmatgiz, M., 1998 | Zbl

[29] Fujita H., “On the blowing up of solutions to the Cauchy problem for $u_t=\Delta u+u^{1+\alpha}$”, J. Fac. Univ. Tokyo. Sect. IA, 13 (1966), 109–124 | MR | Zbl

[30] Levine H. A., “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u)$”, Arch. Rational Mech. and Analys., 51 (1973), 371–386 | MR | Zbl

[31] Amann H., Fila M., “A fujita-type theorem for the laplace equation with a dynamical boundary condition”, Acta Math. Univ. Comenianae, 66:2 (1997), 321–328 | MR | Zbl

[32] Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987 | MR

[33] Rossi J. D., “The blow-up rate for a semilinear parabolic equation with a nonlinear boundary condition”, Acta. Math. Univ. Comenianae, 67:2 (1998), 343–350 | MR | Zbl

[34] Egorov Yu. V., Kondratiev V. A., “On blow-up solutions for parabolic equations of second order”, Differential equations, asymptotic analysis and mathematical physics, Math. Res., 100, Akad. Verlag, Berlin, 1997, 77–84 | MR | Zbl

[35] Mitidieri E., Pokhozhaev S. I., “Apriornye otsenki i otsutstvie reshenii differentsialnykh neravenstv v chastnykh proizvodnykh”, Tr. MI RAN, 232, M., 2001, 1–222 | MR

[36] Laptev G. G., “Ob otsutstvii reshenii odnogo klassa singulyarnykh polulineinykh differentsialnykh neravenstv”, Tr. MI RAN, 232, M., 2001, 223–235 | MR | Zbl

[37] Fridman A., Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, Mir, M., 1968

[38] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1988 | MR

[39] Gilbarg D., Trudinger M., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR | Zbl

[40] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[41] Demidovich B. P., Lektsii po matematicheskoi teorii ustoichivosti, Nauka, M., 1967 | MR