On a higher-order differential inequality involving the Weinstein differential operator in an infinite parallelepiped

We investigate the existence and nonexistence of weak solutions to the higher-order evolution inequality $ \begin{cases} \frac{\partial^k u}{\partial t^k}-\Delta_{3,\gamma} u \ge x_3^{-\xi}|u|^p & \text{in } (0,\infty)\times \Omega,\\[6pt] u(t,x_1,x_2,1) = w(x_1,x_2) & \text{in } (0,\infty)\times \mathbb{R}^2, \end{cases} $ where $ \Omega = \{(x_1,x_2,x_3)\in\mathbb{R}^3:\,01 $, and $ \Delta_{3,\gamma} $ denotes the Weinstein differential operator with parameter $ \gamma\ge-1 $. The boundary datum $ w $ is assumed to be a nonnegative, nontrivial function in $ L^1_{\mathrm{loc}}(\mathbb{R}^2) $. We show that the qualitative behavior of solutions exhibits two distinct regimes depending on the value of $ \gamma $. If $ \gamma\ge1 $, then the existence and nonexistence of weak solutions are separated by the critical value $ \xi^* = 2 $. In contrast, when $ -1\le\gamma2 $, we identify a Fujita-type critical exponent$ p^* = 1+\frac{\xi-2}{1-\gamma}, $which determines the sharp threshold between existence and nonexistence of weak solutions.