Improvement of conditions for finite time blow-up in a fourth-order nonlocal parabolic equation

TL;DR

This paper investigates the blow-up phenomenon in a fourth-order nonlocal parabolic equation, showing that negative initial Nehari functional alone guarantees finite-time blow-up, emphasizing the role of mass conservation.

Contribution

It extends previous results by proving that conditions on initial energy are unnecessary for blow-up, only requiring negative initial Nehari functional.

Findings

Negative initial Nehari functional guarantees finite-time blow-up.

Mass conservation significantly influences solution dynamics.

Previous energy conditions are superfluous for blow-up criteria.

Abstract

This paper is devoted to the study of blow-up phenomenon for a fouth-order nonlocal parabolic equation with Neumann boundary condition, \begin{equation*} \left\{\begin{array}{ll}\ds u_{t}+u_{xxxx}=|u|^{p-1}u-\frac{1}{a}\int_{0}^a|u|^{p-1}u\ dx, & u_x(0)=u_x(a)=u_{xxx}(0)=u_{xxx}(a)=0, & u(x,0)=u_0(x)\in H^2(0, a),\ \ \int_0^au_0(x)\ dx=0, &\end{array}\right. \end{equation*} where $a$ is a positive constant and $p>1$. The existing results on the problem suggest that the weak solution will blow up in finite time if $I(u_0)<0$ and the initial energy satisfies some appropriate assumptions, here $I(u_0)$ is the initial Nehari functional. In this paper, we extend the previous blow-up conditions with proving that those assumptions on the energy functional are superfluous and only $I(u_0)<0$ is sufficient to ensure the weak solution blowing up in finite time. Our conclusion depicts the significant influence of mass conservation on the dynamic behavior of solution.