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

This paper is devoted to the study of blow-up phenomenon for a fouth-order nonlocal parabolic equation with Neumann boundary condition, equation* \array{ll uₓ+uₗₗₗₗ=|u|^p-1u-1a₀ᵃ|u|^p-1u\ dx, & uₓ (0) =uₓ (a) =uₗₗₗ (0) =uₗₗₗ (a) =0, & u (x, 0) =u₀ (x) H² (0, a), \ \ ₀ᵃu₀ (x) \ dx=0, &array. 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 (u₀) <0 and the initial energy satisfies some appropriate assumptions, here I (u₀) 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 (u₀) <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.