Blow‐up phenomenon and the exact blow‐up time for a class of pseudo‐parabolic equations with nonlocal source

Abstract

This paper deals with the blow‐up phenomenon to the following quasi‐linear pseudo‐parabolic equation with nonlocal source: where , is a bounded domain with smooth boundary. Here, 0 < q ≤ p and K(x,y) is an integrable real‐valued function. We show that for q < p, the blow‐up occurs in finite time with suitable initial data and arbitrary positive initial energy. We also state some key results based on the conception of limiting the energy function in the case of nonnegative initial energy. Besides, we obtain the exact blow‐up time under some certain conditions.