Positivity 12 (2008), 75–82
2007 Birkh¨auser Verlag Basel/Switzerland
1385-1292/010075-8, published online October 29, 2007
Gaussian Estimates and Instantaneous Blowup
Jerome A. Goldstein and Ismail Kombe
Helmut H. Schaefer in memoriam
Abstract. For L a second order linear elliptic differential operator on R
is usually interested in ﬁnding positive solutions of the heat equation
= Lu + Vu,
where V is a nonnegative potential. But for L the Laplacian, it was discov-
ered by [BG] in 1984 that positive solutions may not exist if V is too singular.
We use Gaussian estimates to extend this result to the case when L is not
Mathematics Subject Classiﬁcation (2000). 35B30, 35K15, 47D06, 35G10.
Keywords. Instantaneous blowup, Gaussian estimates, heat equation, non-
symmetric elliptic operator, singular potential.
The heat equation
Δu, (x ∈ R
has as its fundamental solution the Gaussian kernel
where σ is a positive constant.
Thus the unique nonnegative solution of (1.1) with initial data u(x, 0) =
f(x) ≥ 0is
(x − y, t)f(y)dy,
provided f ∈ L
), cf. [A2]. Recall that Tychonoﬀ showed in the 1930s that
there are many solutions to (1.1) with ﬁxed initial value f that grow faster than