Self-Similar Blow-Up Profiles for a Reaction-Diffusion Equation with Strong Weighted Reaction

Abstract

We study the self-similar blow-up profiles associated to the following second-order reaction-diffusion equation with strong weighted reaction and unbounded weight:\partial_{t}u=\partial_{xx}(u^{m})+|x|^{\sigma}u^{p},posed for {x\in\mathbb{R}}, {t\geq 0}, where {m>1}, {0<p<1} and {\sigma>\frac{2(1-p)}{m-1}}. As a first outcome, we show that finite time blow-up solutions in self-similar form exist for {m+p>2} and σ in the considered range, a fact that is completely new: in the already studied reaction-diffusion equation without weights there is no finite time blow-up when {p<1}. We moreover prove that, if the condition {m+p>2} is fulfilled, all the self-similar blow-up profiles are compactly supported and there exist two different interface behaviors for solutions of the equation, corresponding to two different interface equations. We classify the self-similar blow-up profiles having both types of interfaces and show that in some cases global blow-up occurs, and in some other cases finite time blow-up occurs only at space infinity. We also show that there is no self-similar solution if {m+p<2}, while the critical range {m+p=2} with {\sigma>2} is postponed to a different work due to significant technical differences.

Eternal solutions for a reaction-diffusion equation with weighted reaction

“…This implies that interfaces cannot exist if m + p < 2. The same dichotomy has been noticed for σ > 2(1 − p)/(m − 1) in [6].…”

“…We notice that there are two different interface behaviors, which are totally different for m+p > 2. These different interface behaviors have been noticed first for the reaction-diffusion equation without weight, that is with σ = 0 (see for example [10]) and more recently these two different interfaces were analyzed in our recent paper [6] in connection with the interface equation, showing that they are essentially different since they satisfy two completely different differential equations for their speed of advance. The presence of these two types of interface is a characteristic of the range 0 < p < 1.…”

“…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.…”

“…(1.5) 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].…”

“…(1.1), 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.…”