[Paper Review] Blow up profiles for a reaction-diffusion equation with critical weighted reaction

Razvan Gabriel Iagar, Ariel Sánchez|arXiv (Cornell University)|Jun 3, 2019

Mathematical and Theoretical Epidemiology and Ecology ModelsMedicine27 references15 citations

TL;DR

This paper classifies self-similar blow-up profiles for a reaction-diffusion equation with a critical weighted reaction term $ u_t = (u^m)_{xx} + |x|^\sigma u^m $, showing that for small $ \sigma > 0 $, multiple compactly supported profiles with interface at a finite point exist, while for large $ \sigma $, no such profiles exist. The key result is a sharp transition in existence due to the weight's influence, with no global-in-space blow-up profiles possible for any $ \sigma > 0 $.

ABSTRACT

We classify the blow up self-similar profiles for the following reaction-diffusion equation with weighted reaction $$ u_t=(u^m)_{xx} + |x|^{\sigma}u^m, $$ posed for $(x,t)\in eal imes(0,T)$, with $m>1$ and $\sigma>0$. In strong contrast with the well-studied equation without the weight (that is $\sigma=0$), on the one hand we show that for $\sigma>0$ sufficiently small there exist \emph{multiple self-similar profiles with interface} at a finite point, more precisely, given any positive integer $k$, there exists $\delta_k>0$ such that for $\sigma\in(0,\delta_k)$, there are at least $k$ different blow up profiles with compact support and interface at a positive point. On the other hand, we also show that for $\sigma$ sufficiently large, the blow up self-similar profiles with interface \emph{cease to exist}. This unexpected balance between existence of multiple solutions and non-existence of any, when $\sigma>0$ increases, is due to the effect of the presence of the weight $|x|^{\sigma}$, whose influence is the main goal of our study. We also show that for any $\sigma>0$, there are no blow up profiles supported in the whole space, that is with $u(x,t)>0$ for any $x\in eal$ and $t\in(0,T)$.

Motivation & Objective

• Understand how the unbounded weight $ |x|^\sigma $ alters blow-up behavior in reaction-diffusion equations with $ m = p $, contrasting with the non-weighted case.
• Classify self-similar blow-up profiles for the equation $ u_t = (u^m)_{xx} + |x|^\sigma u^m $, particularly those with compact support and interface at a finite point.
• Establish the existence and non-existence of good profiles with interface depending on the parameter $ \sigma $, identifying a critical threshold.
• Prove that no blow-up profiles exist that are positive on the entire real line for any $ \sigma > 0 $, ruling out global support.
• Analyze the phase space structure of the associated ODE to determine the behavior of solutions near $ \xi = 0 $ and at interface points.

Proposed method

• Use self-similar solutions of the form $ u(x,t) = (T - t)^{-\alpha} f(\xi) $, with $ \xi = |x|(T - t)^\beta $, to reduce the PDE to an ODE for the profile $ f $.
• Derive the ODE $ (f^m)''(\xi) - \frac{1}{m-1}f(\xi) + \xi^\sigma f^m(\xi) = 0 $, which governs the self-similar profiles.
• Apply phase space analysis to the equivalent first-order system to study the behavior of solutions near $ \xi = 0 $ and at interface points where $ f(\eta) = (f^m)'(\eta) = 0 $.
• Use comparison arguments and energy-type estimates to bound the first intersection points of solution curves with the hyperbola $ (m-1)Y + \sigma X = 0 $ in phase space.
• Perform forward and backward shooting from the origin and from the interface point, respectively, to estimate the range of $ \xi $ where profiles can connect.
• Establish a quantitative threshold $ \sigma > 2m\sqrt{2m/(2m+1)} $ beyond which the forward and backward shooting trajectories cannot meet, proving non-existence of interface profiles.

Experimental results

Research questions

• RQ1 How does the presence of the weight $ |x|^\sigma $ affect the existence and multiplicity of self-similar blow-up profiles with compact support and interface at a finite point?
• RQ2 What is the critical value of $ \sigma $ beyond which no self-similar profiles with interface exist, and how does this threshold depend on $ m $?
• RQ3 Why does the number of such profiles increase with decreasing $ \sigma $, and what mechanism underlies this multiplicity?
• RQ4 Can self-similar blow-up profiles exist that are positive everywhere on $ \mathbb{R} $, and if not, why not?
• RQ5 How does the phase space structure of the ODE governing the profile $ f $ change with varying $ \sigma $, and what does this imply for solution behavior?

Key findings

• For any $ \sigma > 0 $, there are no self-similar blow-up profiles that are positive on the entire real line, meaning no global-in-space blow-up profiles exist.
• For $ \sigma $ sufficiently small, specifically $ \sigma \in (0, \delta_k) $ for any given $ k \in \mathbb{N} $, there exist at least $ k $ distinct self-similar profiles with compact support and interface at a positive point.
• For $ \sigma > 2m\sqrt{2m/(2m+1)} $, no self-similar profiles with interface exist, indicating a sharp transition in solution structure due to the weight.
• The first intersection point of the profile with the critical hyperbola in phase space occurs at $ \xi \leq \xi_+ = \left[ \frac{4m^2(m+1)}{(2m+1)(m-1)^3} \right]^{1/(\sigma+2)} $ when shooting from the origin.
• The last (in backward time) intersection with the hyperbola occurs at $ \xi \geq \xi_- = \left[ \frac{(m+1)\sigma^2}{2m(m-1)^3} \right]^{1/(\sigma+2)} $ when shooting from the interface point.
• When $ \xi_- > \xi_+ $, which occurs for $ \sigma > 2m\sqrt{2m/(2m+1)} $, the forward and backward trajectories cannot meet, proving non-existence of interface profiles for large $ \sigma $.