Blow-up versus extinction in a nonlocal p-Laplace equation with Neumann boundary conditions

This paper studies a fast diffusive p-Laplace equation with the nonlocal source | u | q − ⨍ Ω | u | q d x in a bounded domain, subject to homogeneous Neumann boundary value condition. A critical criterion is determined that the changing sign solutions blow up in finite time with q > 1 and non-positive initial energy associated, and must be global for any initial energy if q ⩽ 1 . In particular, the conditions are obtained under which the changing sign solutions vanish in finite time.