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

Pinsky, Ross

doi:10.1090/S0002-9939-99-05164-3

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

Proc. Amer. Math. Soc. 127 (1999), 3319-3327 Request permission

Abstract:

We consider the inhomogeneous equation \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 $a,\phi \gneqq 0$, $\lambda >0$ 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.

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