On a Dini type blow-up condition for nonlinear higher order differential inequalities
A. A. Kon'kov, A. E. Shishkov
TL;DR
This work establishes a Dini-type blow-up condition ensuring nonexistence of global weak solutions for a high-order differential inequality $\sum_{|\alpha|=m} ∂^α a_α(x,u) \ge g(|u|)$ in $\mathbb{R}^n$, under Carathéodory bounds on the coefficients and a convex, nondecreasing nonlinearity $g$. The main theorem requires $n>m$ and two integrability/ divergence criteria on $g$, namely $\int_1^\infty g^{-1/m}(\zeta)\zeta^{1/m-1}\,d\zeta<\infty$ and $\int_0^1 g(r) r^{-(1+n/(n-m))}\,dr=\infty$, and yields triviality of all global weak solutions; the result also extends to $n\le m$ via previous work. Applications to power-type nonlinearities recover the Ni–Serrin thresholds for the exponent $\lambda$ (and their logarithmic refinements) in the critical regime, demonstrating sharp blow-up behavior. The proof uses an energy functional $E(r)=\int_{B_r} g(|u|)\,dx$, derives a dyadic inequality for $E$, and employs a contradiction argument based on the growth encoded by the Dini-type integrals. Overall, the paper advances the understanding of nonexistence criteria for nonlinear higher-order inequalities and provides sharp blow-up conditions linking the nonlinearity $g$ to the spatial dimension and operator order.
Abstract
We obtain a Dini type blow-up condition for global weak solutions of the differential inequality $$ \sum_{|α| = m} \partial^α a_α(x, u) \ge g (|u|) \quad \mbox{in } {\mathbb R}^n, $$ where $m, n \ge 1$ are integers and $a_α$ and $g$ are some functions.