Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia8341Research 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 решений