A Liouville theorem for vector valued semilinear heat equations with no gradient structure and applications to blow-up

Abstract

Abstract. We prove a Liouville theorem for a vector valued semilinear heat equation with no gradient structure. Classical tools such as the maximum principle or energy techniques break down and have to be replaced by a new approach. We then derive from this theorem uniform estimates for blow-up solutions of that equation.

Construction of a Blow-Up Solution for the Complex Ginzburg–Landau Equation in a Critical Case

“…However, the profile they get is different from the case of the standard heat equation, in the sense that they obtain blow-up for the mode and also for the phase, with a different constant in the profile (see b sub (6)). In the critical case (β = 0, p = δ 2 ), there is no self-similar solution apart from the trivial constant solution w ≡ κ of (14) (see Proposition 2.1 in [NZ10] or simply multiply equation (14) by wρ, where…”

A Liouville theorem for a heat equation and applications for quenching

“…In [MZ98a] and [MZ00], the authors shows that for some a 0 ∈ R and s 0 ∈ R, w a 0 (y, s) = w(y + a 0 e s 0 , s 0 ) satisfies one of the blow-up criteria stated in page (5), which contradicts the fact that w exists for all s ∈ R. In our case we don't have any blow-up criterion. It turns out that this is the major difficulty in our paper, as in [NZ10] for equation (19). Following [NZ10], we will use a geometrical method where the key idea is to extend the convergence stated in (ii) and (iii) from compact sets to larger zones, so that we find the singular profile for w. It appears that in both cases, for larger |y|, this profile becomes strictly inferior to 1 M , where M is defined in (11), which is a contradiction.…”

“…It turns out that this is the major difficulty in our paper, as in [NZ10] for equation (19). Following [NZ10], we will use a geometrical method where the key idea is to extend the convergence stated in (ii) and (iii) from compact sets to larger zones, so that we find the singular profile for w. It appears that in both cases, for larger |y|, this profile becomes strictly inferior to 1 M , where M is defined in (11), which is a contradiction. The originality of our paper is based on Velázquez's work in [Vel92], where he extends the convergence from compact sets to larger sets to find the profile for solutions of (17).…”

“…In this part, we will proceed like in Step 4 in [NZ10]. The following proposition allows us to reach a contradiction.…”

“…Lemma 2.10 (Lemma 3.6 of [NZ10]) Consider I(s) a positive C 1 function such that (50) is satisfied and 0 ≤ I(s) ≤ Ce 3/2s for all s ≤ s 2 , for some s 2 . Then, for some s 3 ≤ s 2 , we have I(s) = 0 for all s ≤ s 3 .…”