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

Considered herein is the initial-boundary value problem for a semilinear parabolic equation with a memory term and non-local source w(t) -delta(B)w -delta(B)w(t) + integral(t)(0) g(t - tau)delta(B)w(tau)d tau = |w| (p-1)w - 1/|B| integral(B) |w|(p-1)w dx(1)/x(1 )dx ' on a manifold with conical singularity, where the Fuchsian type Laplace operator delta(B) is an asymmetry elliptic operator with conical degeneration on the boundary x(1) = 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 w(0) is an element of W and Pi(w(0)) > 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 w(0) is an element of V and Pi(w(0)) < 0, the nonexistence of global solutions, i.e., the solutions blow up in finite time, is obtained by using the convexity technique.