Summary (translated from the Russian): "We prove the nonexistence of global solutions to the quasilinear backward parabolic inequality u_t+{rm div}(|x|^alpha|u|^beta|Du|^{p-2}Du)geq|x|^gamma|u|^{q-1}u, quad xinOmega, t>0, with homogeneous Dirichlet boundary condition and bounded integrable sign-changing initial data, where Omega is a bounded smooth domain in {Bbb R}^N.par "The proof is based on obtaining a priori estimates for the solutions by means of an algebraic analysis of the integral form of the inequality with optimal choice of test functions. We establish conditions for the nonexistence of solutions based on a weak formulation of the problem with test functions of the form phi_{R, epsilon}(x, t)=(pm u^{pm}(x, t)+epsilon)^{delta}varphi_R(x, t) for epsilon>0, delta>0, where u^+ and u^- are the positive and negative parts of the solution u of the problem, and varphi_R is the standard cut-off function whose support depends on the parameter R."