Nonexistence of Global Solutions for Quasilinear Backward Parabolic Inequalities with $p$-Laplace-Type Operator

In this paper, we prove the nonexistence of global solutions to the quasilinear backward parabolic inequality
$$ u_{t}+\operatorname{div}(|x|^{\alpha}|u|^{\beta}|Du|^{p-2}Du) \ge |x|^{\gamma}|u|^{q-1}u,\qquad x\in\Omega,\quad t\ge 0 $$
with homogeneous Dirichlet boundary condition and bounded integrable sign-changing initial function, where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N$. The proof is based on the derivation of a priori estimates for the solutions and involves the algebraic analysis of the integral form of the inequality with an optimal choice of test functions. We establish conditions for the nonexistence of solutions based on the weak formulation of the problem with test functions of the form
$$ \phi_{R,\epsilon}(x,t)=(\pm u^{\pm}(x,t)+\epsilon)^{\delta} \varphi_{R}(x,t)\qquad\text{for}\quad \epsilon>0,\quad \delta>0, $$
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 $R$.