On blow-up conditions for nonlinear higher order evolution inequalities

TL;DR

This paper establishes precise conditions on the nonlinear function in higher order evolution inequalities that determine whether solutions exist globally or blow up, extending understanding of nonlinear PDE behavior.

Contribution

The paper provides exact criteria on the nonlinear term f that ensure all global weak solutions are trivial, advancing the theory of blow-up conditions for higher order PDEs.

Findings

Derived explicit blow-up conditions for solutions

Identified critical growth rates of nonlinear functions

Extended previous results to higher order inequalities

Abstract

For the problem $$ \left\{ \begin{aligned} & \partial_t^k u - \sum_{|\alpha| = m} \partial^\alpha a_\alpha (x, t, u) \ge f (|u|) \quad \mbox{in } {\mathbb R}_+^{n+1} = {\mathbb R}^n \times (0, \infty), & u (x, 0) = u_0 (x), \: \partial_t u (x, 0) = u_1 (x), \ldots, \partial_t^{k-1} u (x, 0) = u_{k-1} (x) \ge 0, \end{aligned} \right. $$ we obtain exact conditions on the function $f$ guaranteeing that any global weak solution is identically zero.