Finite time blow-up for the inhomogeneous equation $u_{t}=\Delta u+a(x)u^{p}+\lambda \phi$ in $R^{d}$

We consider the inhomogeneous equation

where $a,\phi \gneqq 0$, $\lambda >0$ and $f\ge 0$, and give criteria on $p,d,a$, and $\phi$ which determine whether for all $\lambda$ and all $f$ the solution blows up in finite time or whether for $\lambda$ and $f$ sufficiently small, the solution exists for all time.