Geodesic
Behavior of solutions to an inverse problem for a quasilinear parabolic equation
Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1393-1409.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this article we consider the inverse problem with an integral condition by redefinition for a parabolic type equation. The existence of a weak solution of the inverse problem is proved by the Galerkin method.In a bounded domain with a homogeneous Dirichlet condition, sufficient conditions for the destruction of its solution in a finite time are obtained, and also the stability of the solution for the inverse problem with the opposite sign on the nonlinearity of the power type.
Keywords: inverse problems, blowing-up solutions, stability, integral overdetermination condition. 
@article{SEMR_2019_16_a99,
     author = {S. E. Aitzhanov and D. T. Zhanuzakova},
     title = {Behavior of solutions to an inverse problem for a quasilinear parabolic equation},
     journal = {Sibirskie \`elektronnye matemati\v{c}eskie izvesti\^a},
     pages = {1393--1409},
     publisher = {mathdoc},
     volume = {16},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SEMR_2019_16_a99/}
}
TY  - JOUR
AU  - S. E. Aitzhanov
AU  - D. T. Zhanuzakova
TI  - Behavior of solutions to an inverse problem for a quasilinear parabolic equation
JO  - Sibirskie èlektronnye matematičeskie izvestiâ
PY  - 2019
SP  - 1393
EP  - 1409
VL  - 16
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SEMR_2019_16_a99/
LA  - en
ID  - SEMR_2019_16_a99
ER  - 
%0 Journal Article
%A S. E. Aitzhanov
%A D. T. Zhanuzakova
%T Behavior of solutions to an inverse problem for a quasilinear parabolic equation
%J Sibirskie èlektronnye matematičeskie izvestiâ
%D 2019
%P 1393-1409
%V 16
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SEMR_2019_16_a99/
%G en
%F SEMR_2019_16_a99
S. E. Aitzhanov; D. T. Zhanuzakova. Behavior of solutions to an inverse problem for a quasilinear parabolic equation. Sibirskie èlektronnye matematičeskie izvestiâ, Tome 16 (2019), pp. 1393-1409. http://geodesic.mathdoc.fr/item/SEMR_2019_16_a99/

[1] Zh.-L. Lions, Quelques méthodes de résolution des probl`emes aux limites non linéaires, Dunod de Gruyter, Paris, 1969 | MR | Zbl

[2] P.A. Raviart, “Sur la résolution de certaines équations paraboliques non linéaires”, Functional Analysis, 5:2 (1970), 299–328 | MR | Zbl

[3] A.V. Ivanov, “Quasilinear parabolic equations admitting double degeneracy”, St. Petersburg Math. J., 4:6 (1993), 1153–1168 | MR | Zbl

[4] G.I. Laptev, “Weak solutions of second-order quasilinear parabolic equations with double non-linearity”, Sb. Math., 188:9 (1997), 1343–1370 | MR | MR | Zbl

[5] A. Bamberger, “Etude d'une equation doublement non lineaire”, J. Funct. Anal., 24:2 (1997), 148–155 | MR

[6] F. Kh. Mukminov, E. R. Andriyanova, “Stabilization of the solution of a doubly nonlinear parabolic equation”, Sb. math., 204:9 (2013), 1239–1263 | MR | Zbl

[7] M.O. Korpusov, “Destruction of solutions of the heat equation with a double nonlinearity”, Theoret. and Math. Phys., 172:3 (2012), 1173–1176 | MR | Zbl

[8] M.O. Korpusov, “Solution blow-up for a class of parabolic equations with double nonlinearity”, Sb. Math., 204:3 (2013), 323–346 | MR | MR | Zbl

[9] H.A. Levine, S.R. Park, J. Serrin, “Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type”, J. Differential Equations, 142:1 (1998), 212–229 | MR | Zbl

[10] A. Samarskii, V. Galaktionov, S. Kurdyumov, A. Mikhailov, Blow-up in quasilinear parabolic equations, De Gruyter, Berlin, 1995 | MR | Zbl

[11] A.B. Al'shin, M.O. Korpusov, A.G. Sveshnikov, Blow-up in nonlinear Sobolev type equations, Gruyter Ser. Nonlinear Anal. Appl., 15, de Gruyter, Berlin, 2011 | MR | Zbl

[12] A.I. Prilepko, D.G. Orlovsky, I.A. Vasin, Method for Solving Inverse Problems in Mathematical Physics, Monograths and Textbooks in Pure and Applied Mathematics, Marcel Dekker, 2000 | MR

[13] A. G. Ramm, Inverse problems, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2005 | MR | Zbl

[14] R. Riganti, E. Savateev, “Solution of an inverse problem for the nonlinear heat equation”, Commun. Partial Diff. Equations, 19:1 (1994), 1611–1628 | MR | Zbl

[15] I.A. Vasin, “On the existence and uniqueness of the generalized solution of the inverse problem for the nonlinear non stationary Navier-Stokes system in the case of integral overdefinition”, Math. Notes Phys., 54:4 (1993), 1002–1009 | MR | MR | Zbl

[16] A.I. Prilepko, I.A. Vasin, “Investigation of the uniqueness of solutions in some nonlinear inverse problems of hydrodynamics”, Differential equations, 26:1 (1990), 109–120 (in Russian) | MR | Zbl

[17] U.U. Abylkayrov, S.E. Aitzhanov, “Inverse problem for non-stationary system of magnetohydrodynamics”, Boundary Value Problems, 2015:1 (2015), 173 | DOI | MR

[18] S.N. Antontsev, S.E. Aitzhanov, “Inverse problem for an equation with a nonstandard growth condition”, Journal of Applied Mechanics and Technical Physics, 2:60 (2019), 265–277 | MR | MR | Zbl

[19] A. Eden, V.K. Kalantarov, “On global nonexistence of solutions to an inverse problem for semilinear parabolic equations”, J. Math. Anal. Appl., 307 (2005), 120–133 | MR | Zbl

[20] S. Gur, M. Yaman, Y. Yilmaz, “Finite time blow up of solutions to an inverse problem for a quasilinear parabolic equation with power nonlinearity”, J. Nonlinear Sci. Appl., 9 (2016), 1902–1910 | MR | Zbl

[21] J. Soviet Math., 10:1 (1978), 53–70 | MR | MR