Geodesic
Solvability and Blow-Up of Weak Solutions of Cauchy Problems for $(3+1)$-Dimensional Equations of Drift Waves in a Plasma
Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 459-475.

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In this paper, two Cauchy problems that contain different nonlinearities $|u|^q$ and $(\partial/\partial t)|u|^q$ are studied. The differential operator in these problems is the same. It is defined by the formula $\mathfrak{M}_{x,t}:=(\partial^2/\partial t^2)\Delta_{\perp}+ \partial^2/\partial x_3^2$. The problems have a concrete physical meaning, namely, they describe drift waves in a magnetically active plasma. Conditions are found under which weak generalized solutions of these Cauchy problems exist and also under which weak solutions of the same Cauchy problems blow up. However, the question of the uniqueness of weak generalized solutions of Cauchy problems remains open, because uniqueness conditions have not been found.
Keywords: Sobolev-type nonlinear equations, blow-up, local solvability, nonlinear capacity. 
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     title = {Solvability and {Blow-Up} of {Weak} {Solutions} of {Cauchy} {Problems} for $(3+1)${-Dimensional} {Equations} of {Drift} {Waves} in a {Plasma}},
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R. S. Shafir. Solvability and Blow-Up of Weak Solutions of Cauchy Problems for $(3+1)$-Dimensional Equations of Drift Waves in a Plasma. Matematičeskie zametki, Tome 111 (2022) no. 3, pp. 459-475. http://geodesic.mathdoc.fr/item/MZM_2022_111_3_a12/

[1] M. O. Korpusov, R. S. Shafir, “O razrushenii slabykh reshenii zadachi Koshi dlya $3+1$-mernogo uravneniya dreifovykh voln v plazme”, Zh. vychisl. matem. i matem. fiz., 62:1 (2022), 124–158 | DOI

[2] A. B. Al'shin, M. O. Korpusov, A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter Ser. Nonlinear Anal. Appl., 15, Walter de Gruyter, Berlin, 2011 | MR

[3] G. A. Sviridyuk, “K obschei teorii polugrupp operatorov”, UMN, 49:4 (298) (1994), 47–74 | MR | Zbl

[4] S. A. Zagrebina, “Nachalno-konechnaya zadacha dlya uravnenii sobolevskogo tipa s silno $(L,p)$-radialnym operatorom”, Matem. zametki YaGU, 19:2 (2012), 39–48

[5] A. A. Zamyshlyaeva, G. A. Sviridyuk, “Nonclassical equations of mathematical physics. Linear Sobolev type equations of higher order”, Vestn. Yuzhno-Ur. un-ta. Ser. Matem. Mekh. Fiz., 8:4 (2016), 5–16 | DOI | MR | Zbl

[6] B. V. Kapitonov, “Teoriya potentsiala dlya uravneniya malykh kolebanii vraschayuscheisya zhidkosti”, Matem. sb., 109 (151):4 (8) (1979), 607–628 | MR | Zbl

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

[8] S. A. Gabov, Novye zadachi matematicheskoi teorii voln, Fizmatlit, M., 1998

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

[10] S. I. Pokhozhaev, E. Mitidieri, “Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh”, Trudy MIAN, 234, Nauka, MAIK «Nauka/Interperiodika», M., 2001, 3–383 | MR | Zbl

[11] E. I. Galakhov, “Some nonexistence results for quasilinear elliptic problems”, J. Math. Anal. Appl., 252:1 (2000), 256–277 | DOI | MR | Zbl

[12] E. I. Galakhov, O. A. Salieva, “Ob otsutstvii neotritsatelnykh monotonnykh reshenii dlya nekotorykh koertsitivnykh neravenstv v poluprostranstve”, Differentsialnye i funktsionalno-differentsialnye uravneniya, SMFN, 63, no. 4, Rossiiskii universitet druzhby narodov, M., 2017, 573–585 | DOI

[13] M. O. Korpusov, “Kriticheskie pokazateli mgnovennogo razrusheniya ili lokalnoi razreshimosti nelineinykh uravnenii sobolevskogo tipa”, Izv. RAN. Ser. matem., 79:5 (2015), 103–162 | DOI | MR | Zbl

[14] M. O. Korpusov, “O razrushenii reshenii nelineinykh uravnenii tipa uravneniya Khokhlova–Zabolotskoi”, TMF, 194:3 (2018), 403–417 | DOI | MR | Zbl

[15] M. O. Korpusov, A. V. Ovchinnikov, A. A. Panin, “Instantaneous blow-up versus local solvability of solutions to the Cauchy problem for the equation of a semiconductor in a magnetic field”, Math. Methods Appl. Sci., 41:17 (2018), 8070–8099 | DOI | MR | Zbl

[16] M. O. Korpusov, Yu. D. Pletner, A. G. Sveshnikov, “O nestatsionarnykh volnakh v sredakh s anizotropnoi dispersiei”, Zh. vychisl. matem. i matem. fiz., 39:6 (1999), 1006–1022 | MR | Zbl

[17] V. R. Kudashev, A. B. Mikhailovskii, S. E. Sharapov, “K nelineinoi teorii dreifovoi mody, indutsirovannoi toroidalnostyu”, Fizika plazmy, 13:4 (1987), 417–421

[18] F. F. Kamenets, V. P. Lakhin, A. B. Mikhailovskii, “Nelineinye elektronnye gradientnye volny”, Fizika plazmy, 13:4 (1987), 412–416

[19] A. P. Sitenko, P. P. Sosenko, “O korotkovolnovoi konvektivnoi turbulentnosti i anomalnoi elektronnoi teploprovodnosti plazmy”, Fizika plazmy, 13:4 (1987), 456–462

[20] A. A. Panin, “O lokalnoi razreshimosti i razrushenii resheniya abstraktnogo nelineinogo integralnogo uravneniya Volterra”, Matem. zametki, 97:6 (2015), 884–903 | DOI | MR | Zbl