Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)8341Research ArticleNonexistence of Global Solutions of Quasi-linear Backward Parabolic EquationsTsegawB BDepartment of Mathematical Analysis and Theory of Functionsbirilewb@yahoo.comPeoples’ Friendship University of Russia150220142112608092016Copyright © 2014,2014This paper deals with the nonexistence of global solutions of quasi-linear backward parabolic equations for p-Laplacian operators: ut = −div Dup−2Du + uq−1u, x,t ∈ Ω × (0,∞) with the Dirichlet boundary condition u = 0 on the boundary ∂Ω × (0,∞) and a bounded integrable initial function u(x,0) = u0(x), where Ω is a smoothly bounded domain in ℝN. We also consider this problem in the case of Ω = ℝN. The problem is analyzed using the test function method, developed by E. L. Mitidieri and S. I. Pohozaev [Mitidieri E., Pohozaev S. I. A Priory Estimates and the Absence of Solutions of Non- linear Partial Diﬀerential Equations and Inequalities // Proceedings of the Steklov Institute of Mathematics, 2001. - Vol. 234, No 3. - 362 p. - (in russian).] It is based on deriving a priory estimates for solutions by an algebraic analysis of the integral form of inequalities with an optimal choice of test functions. With the help of this method, we obtain the nonexistence conditions based on the weak formulation of the problem with test functions of the form: φ(x,t) = ±u±(x,t) + εδφR(x,t),for ε 0,δ 0, where u+ and u− are the positive and negative parts of the solution u of the problem respectively and φR is a standard cut-oﬀ function whose support depends on a parameter R 0.Quasi-linear backward parabolic equationsp-Laplacian operatorstest function methodapriori estimates and nonexistence of global solutionsквазилинейные обратные параболические уравненияоператор p-Лапласаметод пробных функцийаприорные оценки и отсутствие глобальных решенийМитидиери Э., Похожаев С. И. Априорные оценки и отсутствие решений нелинейных уравнений и неравенств в частных производных // Тр. МИАН. - М.: Наука, 2001. - Т. 234, № 3. - 362 с.Gong-ming W., Zu-chi C. Nonexistence of Global Positive Solutions for Quasi-Linear Parabolic Inequalities // Jou. Univ. Sc. & Tech. China. - 2004. - Vol. 34, No 3. - Pp. 2903-2916.Ma L. Blow - Up for Semi-Linear Parabolic Equations with Critical Sobolev Exponent // Pure and Applied Analysis. - 2013. - Vol. 12, No 2. - Pp. 1103-1110. - doi:10.3934/cpaa.2013.12.1103.Fila M., Souplet P. The Blow-Up Rate for Semi-Linear Parabolic Problems on General Domains. - Basel: Birkhauser Verlag, 2001. - Vol. 8, Pp. 473-480.Gazzola F., Weth T. Finite Time Blow-Up and Global Solutions for Semilinear Parabolic Equations with Initial Data at High Energy Level // Differential and Integral Equations. - 2005. - Vol. 18, No 9. - Pp. 961-990.Pohozaev S. Blow-Up of Global Sign-Changing Solutions of a Nonlinear Heat Equation // Doklady Mathematics. - 2012. - Vol. 85, issue 2. - Pp. 225-228.Polacik P., Quittner P. A Liouville-Type Theorem and the Decay of Radial Solutions of a Semilinear Heat Equation // Nonlinear Anal. - 2006. - Vol. 64. - Pp. 1679-1689.Souplet P. An Optimal Liouville-Type Theorem for Radial Entire Solutions of the Porous Medium Equation with Source // J. Differential Equations. - 2009. - Vol. 246. - Pp. 3980 - 4005.Galaktionov V., Vazquez J. L. Extinction for a Quazilinear Heat Equation with Absorption // Comm. PDE. - 1994. - Vol. 19. - Pp. 1075-1106.Caristi G., Mitidieri E. Existence and Nonexistence of Global Solutions of Higher-Order Parabolic Problem with Slow Decay Initial Data // J. Math. Anal. Appl. - 2003. - Vol. 279. - Pp. 710-722.Gazzola F., Grunau H. C. Global Solutions for Superlinear Parabolic Equations Involving the Biharmonic Operator for Intial Data with Optimal Slow Decay // Calculus of Variations. - 2007. - Vol. 30. - Pp. 389-415.Sun F. Life Span of Blow-Up Solutions for Higher-Order Semilinear Parabolic Equations // Electronic Journal of Differential Equations. - 2010. - Vol. 17. - Pp. 1-9. - http://ejde.math.txstate.edu.Guedda M., Kirane M. Local and Global Nonexistence of Solutions to Semilinear Evolution Equations // Electronic Journal of Differential Equations. - 2002. - Vol. 9. - Pp. 149-160. - http://ejde.math.swt.edu.G. Cai H. P., Xing R. A Note on Parabolic Liouville Theorems and Blow-Up Rates for a Higher-Order Semilinear Parabolic System // International Journal of Differential Equations. - 2011. - Vol. 2011. - Pp. 1-9. - doi:10.1155/2011/ 896427.Zubelevich O. On Weakly Nonlinear Backward Parabolic Problem // International Journal of Differential Equations. - 2009. - Pp. 9-28. - http://www.ma.utexas.edu/mp_arc/c/09/09-28.pdf.