Instantaneous blow up

The Hardy inequality and nonlinear parabolic equations on Carnot groups

“…These inequalities play an important role in the study of linear and nonlinear partial differential equations on Euclidean space R n and on the Heisenberg group H n (see [1], [4], [7], [8], [11], [16], [19], [20], [21], [22], [23], [26], [27], [33]). …”

“…The observation of Cabré and Martel [8] used to get additional results on nonexistence of positive solutions for wide class of linear and nonlinear parabolic problems on Euclidean space R n and the Heisenberg group H n ( see [18], [19], [20], [21], [23], [26], and [27]). The purpose of this paper is to investigate nonexistence of positive solutions of the problem(1.1) on more general Carnot group G and our results recover the previous results [20], [21].…”

Self-similar asymptotics of solutions to heat equation with inverse square potential

“…The initial value problem (1.1)-(1.2) was popularized by Baras and Goldstein [1] who discovered the "instantaneous blow up" of solutions, namely, the fact that Cauchy problem (1.1)-(1.2) has no positive local in time solutions if λ > (n−2) 2 4 (see also [7] for a simple proof via the Harnack inequality). Moreover, for 0 < λ (n−2) 2 4 , the authors of [1] found necessary and sufficient conditions for u 0 so that a nonnegative solution exists.…”

The linear heat equation with highly oscillating potential

“…[3], [8], [9] and [6]. The motivating work for these papers is the classical paper of Baras and Goldstein [2], in which it was shown that the initial value problem 2 ) 2 .…”

“…Remark 3.1. In the paper [6], Goldstein and Kombe conjectured that if c decreases to below C * (N ) and |β| is large enough, then there is instantaneous blow-up. We now see that in the light of Theorem 2.3, this is not so.…”