Geodesic
On the blowup of solutions of the Cauchy problem for nonlinear equations of ferroelectricity theory
Teoretičeskaâ i matematičeskaâ fizika, Tome 212 (2022) no. 3, pp. 327-339 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study two Cauchy problems for nonlinear equations of the Sobolev type, of the form $ \frac{\partial}{\partial t}\frac{\partial^2u}{\partial x_3^2} + \Delta u=|u|^q $ and $ \frac{\partial}{\partial t}\Delta_{\perp}u + \Delta u= |u|^q$. We find conditions under which weak generalized local-in-time solutions of the Cauchy problem exist, and we also find conditions under which solutions blow up.
Keywords: Sobolev-type nonlinear equations, blowup, local solvability, nonlinear capacity. 
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M. O. Korpusov; R. S. Shafir. On the blowup of solutions of the Cauchy problem for nonlinear equations of ferroelectricity theory. Teoretičeskaâ i matematičeskaâ fizika, Tome 212 (2022) no. 3, pp. 327-339. http://geodesic.mathdoc.fr/item/TMF_2022_212_3_a0/

[1] M. O. Korpusov, R. S. Shafir, “O razrushenii slabykh reshenii zadachi Koshi dlya $(3+\nobreak1)$-mernogo uravneniya dreifovykh voln v plazme”, Zh. vychisl. matem. i matem. fiz., 62:1 (2022), 124–158 | DOI | DOI

[2] A. B. Al'shin, M. O. Korpusov, A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter Series in Nonlinear Analysis and Applications, 15, De Gruyter, Berlin, New York, 2011 | DOI | MR

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

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

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

[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] E. Mitidieri, S. I. Pokhozhaev, “Apriornye otsenki i otsutstvie reshenii nelineinykh uravnenii i neravenstv v chastnykh proizvodnykh”, Trudy MIAN, 234 (2001), 3–383 | MR | Zbl

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

[12] E. I. Galakhov, O. A. Salieva, “Ob otsutstvii neotritsatelnykh monotonnykh reshenii dlya nekotorykh koertsitivnykh neravenstv v poluprostranstve”, Sovremennaya matematika. Fundamentalnye napravleniya, 63:4 (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 | DOI | MR

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

[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

[16] V. A. Ditkin, A. P. Prudnikov, Spravochnik po operatsionnomu ischisleniyu, Vysshaya shkola, M., 1965 | Zbl