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

Abstract

This paper deals with a class of nonlocal semilinear pseudo-parabolic equation with conical degenerationut−△But−△Bu=|u|p−1u−1|B|∫B|u|p−1udx1x1dx′, on a manifold with conical singularity, where △B is Fuchsian type Laplace operator with totally characteristic degeneracy on the boundary x1=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(u0)<d and critical initial energy J(u0)=d, respectively. Furthermore, we investigate the global existence and finite time blow up of the solutions with the high initial energy J(u0)>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 level and critical initial level.