Construction of a blow-up solution for a perturbed nonlinear heat equation with a gradient and a non-local term

Abstract

We consider in this paper a perturbation of the standard semilinear heat equation by a term involving the space derivative and a non-local term. In some earlier works [1, 2], we constructed a solution u for that equation such that u and ∇u both blow up at the origin and only there. We also gave the final blow-up profile. In this paper, we refine our construction method in order to get a sharper estimate on the gradient at blow-up.

Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term

“…Note that already when µ = 0, the profile in (1.8) is sharper than the profile derived in [20], in the sense that we divide here the bound by 1 + ( |x| 2 T −t ) β 2 . Despite this sharp estimate, we left two questions unanswered in [1]:…”

“…Our argument in [1] is a non trivial adaptation of the pioneering work performed for equation (1.3) by Bricmont and Kupiainen [4] and Merle and Zaag [20] (see also the note [21]). More precisely, the proof is given in the similarity variables setting: we linearize the PDE around the profile candidate f defined in (1.10) (actually, around a small perturbation of f ), then, we control the nonpositive part of the spectrum thanks to the decaying properties of the linear operator.…”

“…where 1 1−u may blow up in finite time (see Duong and Zaag [6] and the references therein). Specific difficulties arise in the study of blow-up for both equations (1.4) and (1.5), as one may see from the constructions of singular solutions in Tayachi and Zaag [26] (see also the note [27]) and Duong and Zaag [6].…”

“…In this paper, following our earlier work [1] on (1.1), we intend to consider more complicated perturbations of equation (1.3). Specifically, since it shows a product between a gradient term and an integral term, we felt that equation (1.1) can be the next challenge to study blow-up for parabolic PDEs.…”

“…In [1], we constructed a solution u(x, t) for equation (1.1) which blows up in finite time T at a = 0, and we proved that the solution behaves as follows: for all (x, t) ∈ R N × [0, T ), (1.9) (note that 0 ≤ β < 1 from (1.7) and (1.2)), where the "intermediate" profile f is given by…”