On the problem of preventing blowing-up and quenching for semilinear heat equation (Q1820927)

The global existence of solutions to the IVP \[ u_ t=\Delta u+g(t)f(u)\quad (t>0),\quad u|_{t=0}=u_ 0(x) \] and the IBVP \[ u_ t=\Delta u+g(t,x)f(u)\quad (t>0,\quad x\in \Omega),\quad u|_{t=0}=u|_{\partial \Omega}=0 \] is investigated. As has been done by \textit{Chen Qingyi} [Acta Math. Sci. 2, 17-23 (1982; Zbl 0517.35049)], the introduction of a factor g(t) or g(t,x) in the nonlinear term in order to prevent the blowing-up or quenching of solutions. It is shown in this paper that most of the restrictions on f,g and \(u_ 0\) in the theorems of the above paper may be cancelled or relaxed, that the smallness of g is required only for t large, and that under certain conditions controlling initial state can avoid blowing-up.