Self-similar Blow-Up Profiles for a Reaction–Diffusion Equation with Critically Strong Weighted Reaction

Eternal solutions for a reaction-diffusion equation with weighted reaction

“…We stress here that in our previous papers [6,7] there were different types of good profiles with interface, in particular profiles starting with f (0) = 0 and satisfying a left-interface condition at ξ = 0, namely (f m ) (0) = 0. The subsequent analysis will show that such profiles do no longer exist in our range of exponents (1.2), this is why we do not consider them in the definition.…”

“…(1.1), there are two facts indicating that eternal solutions, in this case with exponential grow-up in time, might exist. On the one hand, we have shown in two recent papers [6,7] that self-similar blow-up profiles to Eq. (1.1) with m > 1, p ∈ (0, 1) (only in dimension N = 1) exist precisely for σ > 2(1 − p)/(m − 1), and we also classified them.…”

“…(1.5) have to intersect the axis ξ = 0 at some finite point. Another significant difference between the case we are dealing with here and the ranges of exponents studied in our previous works [6,7] is the fact that the self-similar exponents α and β are not fixed : one of them is totally free and is in fact the most important parameter of the problem. This is a situation which is characteristic to the very specific cases where eternal solutions exist, see for example [5].…”

“…(1.5). The dynamical system that we use in the proofs is very different from the ones we used (in three dimensions) in our recent papers [6,7], since that one is no longer satisfactory at a technical level. Thus, we will go back to the variables employed for the dynamical system in the older paper [8].…”

Self-similar blow-up patterns for a reaction-diffusion equation with weighted reaction in general dimension

“…While a detailed analysis of selfsimilarity for the case σ = 0 is available in [32,Chapter 4], previous works on different ranges of the exponents m, p and σ > 0 led to some completely novel and, in many occasions, unexpected forms and behaviors of the profiles, see for example [15] (for p = 1 and σ > 0), [18] (for p ∈ (1, m) and σ > 0) or [16,20] (for p = m and dimensions N = 1, respectively N ≥ 2), all them for the range m > 1. A separate mention has to be given to the works [17,19], dealing with our range of exponents (1.2) but for dimension N = 1, where we classify the self-similar blow-up profiles and obtain that there are two different types of compactly supported profiles, depending on the interface behavior and also on the sign of the number m + p − 2, a fact that is in line with the more qualitative study performed in the series of papers by De Pablo and Vázquez [26,27,28] for the homogeneous equation, that is, Eq. (1.1) with σ = 0.…”