[Paper Review] Optimal condition for blow-up of the critical $L^q$ norm for the semilinear heat equation

Noriko Mizoguchi, Philippe Souplet|arXiv (Cornell University)|Dec 29, 2018

Nonlinear Partial Differential Equations37 references4 citations

TL;DR

This paper establishes that the critical $L^{q^*}$ norm of solutions to the semilinear heat equation $u_t = \Delta u + |u|^{p-1}u$ blows up if and only if the solution undergoes type I blow-up, under general conditions on $p>1$ and smooth domains $\Omega$. The proof combines similarity variables, rescaling, backward uniqueness, and unique continuation for parabolic equations, resolving a long-standing open question by showing this condition is essentially optimal given known type II blow-up solutions with bounded critical norm.

ABSTRACT

We shed light on a long-standing open question for the semilinear heat equation $u_t = Δu + |u|^{p-1} u$. Namely, without any restriction on the exponent $p>1$ nor on the smooth domain~$Ω$, we prove that the critical $L^q$ norm blows up whenever the solution undergoes {\it type~I~blow-up.} A~similar property is also obtained for the local critical $L^q$ norm near any blow-up point. In view of recent results of existence of type~II blow-up solutions with bounded critical $L^q$ norm, which are counter-examples to the open question, our result seems to be essentially the best possible result in general setting. This close connection between type I blow-up and critical $L^q$ norm blow-up appears to be a completely new observation. Our proof is rather involved and requires the combination of various ingredients. It is based on analysis in similarity variables and suitable rescaling arguments, combined with {\it backward uniqueness and unique continuation properties} for parabolic equations. As a by-product, we obtain the nonexistence of self-similar profiles in the critical $L^q$ space. Such properties were up to now only known for $ p \le p_S $ and in radially symmetric case for $ p > p_S $, where $ p_S $ is the Sobolev exponent.

Motivation & Objective

• To resolve the long-standing open question of whether the critical $L^{q^*}$ norm must blow up when solutions to the semilinear heat equation undergo blow-up.
• To establish that type I blow-up is both necessary and essentially sufficient for critical $L^{q^*}$ norm blow-up, under general $p>1$ and smooth domains.
• To show that the result is optimal by connecting it to known counterexamples of type II blow-up with bounded $L^{q^*}$ norm.
• To prove the nonexistence of self-similar profiles in the critical $L^{q^*$ space, extending known results beyond the Sobolev exponent $p_S$.
• To develop a refined analysis in similarity variables and use backward uniqueness and unique continuation to establish the blow-up criterion.

Proposed method

• Transforming the semilinear heat equation into similarity variables via the change $v(s,y) = (T-t)^{\beta} u(t,x)$ with $s = -\log(T-t)$, $y = x/(T-t)^\beta$.
• Applying rescaling arguments to analyze the behavior of solutions near blow-up time, focusing on local $L^{q^*}$ norms near blow-up points.
• Using backward uniqueness for parabolic equations to rule out nontrivial solutions that vanish in a region after blow-up time.
• Employing unique continuation properties (e.g., from [17]) to extend the vanishing of solutions from a region to the whole domain.
• Combining parabolic regularity estimates and uniform bounds in annular regions to derive contradiction from the assumption of bounded $L^{q^*}$ norm under type I blow-up.
• Analyzing the asymptotic behavior of rescaled solutions and using the structure of the critical space $L^{q^*}$ with $q^* = N(p-1)/2$.

Experimental results

Research questions

• RQ1 Does the critical $L^{q^*}$ norm necessarily blow up when a solution to the semilinear heat equation undergoes type I blow-up?
• RQ2 Can type II blow-up solutions exist with uniformly bounded $L^{q^*}$ norm, and if so, what does this imply for the necessity of the $L^{q^*}$ blow-up condition?
• RQ3 Is the condition of type I blow-up both necessary and sufficient for the blow-up of the critical $L^{q^*}$ norm in general domains and for all $p>1$?
• RQ4 What is the role of backward uniqueness and unique continuation in ruling out self-similar profiles in the critical $L^{q^*$ space?
• RQ5 Can the nonexistence of self-similar profiles in $L^{q^*}$ be established beyond the known cases of $p \leq p_S$ or radial symmetry?

Key findings

• The critical $L^{q^*}$ norm blows up if and only if the solution undergoes type I blow-up, for any $p>1$ and smooth domain $\Omega$.
• This result is optimal in the sense that known type II blow-up solutions (e.g., in [46], [12]) have uniformly bounded $L^{q^*}$ norms, showing that type I blow-up is necessary for $L^{q^*}$ blow-up.
• The paper proves the nonexistence of self-similar profiles in the critical $L^{q^*}$ space, extending previous results that were limited to $p \leq p_S$ or radial symmetry.
• The local critical $L^{q^*}$ norm near any blow-up point also blows up under type I blow-up, confirming the local nature of the criterion.
• The proof relies on a novel combination of similarity variables, rescaling, backward uniqueness, and unique continuation, establishing a new connection between blow-up type and $L^{q^*}$ behavior.
• The result holds without restrictions on $p>1$ or domain geometry, making it a general and sharp criterion for $L^{q^*}$ norm blow-up.