On the problem of preventing blowing-up and quenching for semilinear heat equation

In this paper, the global existence of solutions to the IVP =Δu+g(t)ƒ(u) (t>o), u|_t-o=u_o(x) and the PVP u_t=Δu-g(t,x)f(u) (t>0,x∈Ω) is investigated. As he heen done in 6], the in, faction of factor g(t) or g(t, x) in nonlinear term is to prevent the occurrance of blowing-up or quenching of solutions. It is shown in this paper that most of the restrictions onf, g and u0 in the theorems of 6] may be cancelled or relaxed, that the smallness ofg is required only for t large, and that under certain conditions controlling initial state can avoid blowing-up.