Global well-posedness for a nonlocal semilinear pseudo-parabolic equation with conical degeneration

This paper deals with a class of nonlocal semilinear pseudo-parabolic equation with conical degeneration u(t) - Delta(B)u(t) - Delta(B)u = vertical bar u vertical bar(p-1) u-1/vertical bar B vertical bar integral(B) vertical bar u vertical bar(p-1)udx(1)/x(1)dx', on a manifold with conical singularity, where Delta(B) is Fuchsian type Laplace operator with totally characteristic degeneracy on the boundary x(1)= 0. By using the modified method of potential well with Galerkin approximation and concavity, the global existence, uniqueness, finite time blow up and asymptotic behavior of the solutions will be discussed at the low initial energy J(u(0)) < dand critical initial energy J(u(0)) = d, respectively. Furthermore, we investigate the global existence and finite time blow up of the solutions with the high initial energy J(u(0)) > d by the variational method. Especially, we also derive the threshold results of global existence and nonexistence for the solutions at two different initial energy levels, i.e. low initial leveland critical initial level. (c) 2020 Elsevier Inc. All rights reserved.