Finite time blow-up for the inhomogeneous equation 𝑢_{𝑡}=Δ𝑢+𝑎(𝑥)𝑢^{𝑝}+𝜆𝜙 in 𝑅^{𝑑}

We consider the inhomogeneous equation a m p ; u t = Δ u + a ( x ) u p + λ ϕ ( x ) in R d , t ∈ ( 0 , T ) , a m p ; u ( x , 0 ) = f ( x ) , \begin{equation*} \begin {split} & u_{t}=\Delta u+a(x)u^{p}+\lambda \phi (x) \text {in} R^{d}, t\in (0,T),\\ &u(x,0)=f(x),\end{split}\end{equation*} where a , ϕ ≩ 0 a,\phi \gneqq 0 , λ > 0 \lambda >0 and f ≥ 0 f\ge 0 , and give criteria on p , d , a p,d,a , and ϕ \phi which determine whether for all λ \lambda and all f f the solution blows up in finite time or whether for λ \lambda and f f sufficiently small, the solution exists for all time.