Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities |u| _t ≥ −Δ² u + |v| ^q , |v| _t ≥ −Δ² v + |u| ^p in \mathbbS = \mathbbRⁿ ×\mathbbR⁺\mathbb{S} = \mathbb{R}^n \times \mathbb{R}^ + with p, q > 1, n ≥ 1. A Fujita-Liouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on \mathbbS\mathbb{S} whenever $\tfrac{n} {4} \leqslant \max \left( {\tfrac{{p + 1}} {{pq - 1}} - \tfrac{{q + 1}} {{pq - 1}}} \right)$\tfrac{n} {4} \leqslant \max \left( {\tfrac{{p + 1}} {{pq - 1}} - \tfrac{{q + 1}} {{pq - 1}}} \right). Since the general maximum-comparison principle does not hold for the fourth-order problem, the authors use the test function method to get the global non-existence of nontrivial solutions.