Fujita-Liouville type theorem for coupled fourth-order parabolic inequalities

Abstract

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 $\mathbb{S} = \mathbb{R}^n \times \mathbb{R}^ +$ 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 $\mathbb{S}$ whenever $\tfrac{n}{4} \leqslant \max \left( {\tfrac{{p + 1}}{{pq - 1}} - \tfrac{{q + 1}}{{pq - 1}}} \right)$. 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.

Keywords

Fujita exponent / Liouville type theorem / Higher-order parabolic inequalities / Test function method

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