Nonlocal Pseudo-Parabolic Equation with Memory Term and Conical Singularity: Global Existence and Blowup

Abstract

Considered herein is the initial-boundary value problem for a semilinear parabolic equation with a memory term and non-local source wt−ΔBw−ΔBwt+∫0tg(t−τ)ΔBw(τ)dτ=|w|p−1w−1|B|∫B|w|p−1wdx1x1dx′ on a manifold with conical singularity, where the Fuchsian type Laplace operator ΔB is an asymmetry elliptic operator with conical degeneration on the boundary x1=0. Firstly, we discuss the symmetrical structure of invariant sets with the help of potential well theory. Then, the problem can be decomposed into two symmetric cases: if w0∈W and Π(w0)>0, the global existence for the weak solutions will be discussed by a series of energy estimates under some appropriate assumptions on the relaxation function, initial data and the symmetric structure of invariant sets. On the contrary, if w0∈V and Π(w0)<0, the nonexistence of global solutions, i.e., the solutions blow up in finite time, is obtained by using the convexity technique.