Global existence and nonexistence for semilinear parabolic equations with conical degeneration

摘要

In this article we study the initial boundary value problem of semilinear parabolic equations u_t - Δ_(Bu) = |u|~(p-1)u on a manifold with conical singularity, where Δ_B is Fuchsian type Laplace operator investigated in Chen et al. (Calc Var 43:463-184, 2012) with totally characteristic degeneracy on the boundary x = 0. By using a family of potential wells, we obtain existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions. Especially, the relation between the above two phenomena is derived as a sharp condition.