Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance

This paper establishes the non-uniqueness of positive solutions for Cauchy problems of supercritical semilinear heat equations by proving that the existence of a positive radial singular stationary solution guarantees at least two distinct solutions starting from that singular initial data.

The Gist

Here is an explanation of the paper "Non-Uniqueness of Positive Solutions for Supercritical Semilinear Heat Equations Without Scale Invariance" by Kotaro Hisa and Yasuhito Miyamoto, translated into everyday language with creative analogies.

The Big Picture: The "One Initial State, Two Different Futures" Mystery

Imagine you are a weather forecaster. You look at the current temperature map of the world (the initial state). Usually, if you know the current weather perfectly and the laws of physics are fixed, you can predict exactly what the weather will be tomorrow, next week, or next year. In mathematics, this is called uniqueness: one starting point leads to exactly one future path.

However, this paper discovers a fascinating exception. The authors prove that for a specific, extreme type of heat equation (a mathematical model for how heat spreads), if you start with a very specific, "explosive" initial temperature, the future is not determined. The system can split into two completely different realities at the exact same moment.

• Reality A: The temperature stays exactly as it is (a static, singular state).
• Reality B: The temperature immediately starts changing, smoothing out, and evolving over time.

Both realities are mathematically valid, and both start from the exact same starting point.

---

The Cast of Characters

To understand how this happens, let's meet the players in this mathematical drama:

1. The Heat Equation (The Stage)

Think of the heat equation as the rulebook for how heat moves. Usually, heat flows from hot spots to cold spots, smoothing everything out.

• The Twist: This paper deals with "super-critical" heat. Imagine a fire that doesn't just burn; it burns faster the hotter it gets. The fuel (the nonlinearity $f(u)$) is so intense that the heat generation explodes.

2. The Singular Solution ($u^*$) (The Volcano)

The authors start with a very special initial condition called a singular stationary solution.

• The Analogy: Imagine a volcano that has been frozen in time. The center is infinitely hot (a singularity), but the heat isn't spreading or changing; it's perfectly balanced in a static, explosive state. It's a "frozen explosion."
• In the math world, this is a solution that exists forever without changing.

3. The "Normal" Solution ($u(t)$) (The Thawing)

The paper proves that if you start with that "frozen volcano," there is another valid future where the system doesn't stay frozen. Instead, it immediately "thaws." The infinite heat at the center begins to spread out, the temperature drops, and the system evolves into a normal, smooth wave of heat.

---

The Core Discovery: Breaking the Rules

In most physics problems, if you have a "frozen volcano" (the singular solution), you might think it's the only thing that can happen. But these authors showed that for certain types of extreme heat growth (specifically when the growth rate is between two critical thresholds, known as the Sobolev and Joseph-Lundgren exponents), the "frozen volcano" is unstable in a very specific way.

The Metaphor of the Tightrope:
Imagine a tightrope walker standing perfectly still on a wire.

• Standard Physics: If you nudge them, they fall. If they are perfectly balanced, they stay there.
• This Paper's Discovery: The authors found a scenario where the tightrope walker can stand perfectly still (Solution A), but at the exact same time, they can also decide to start walking forward (Solution B). Both actions are physically possible starting from that exact spot. The universe doesn't force a choice; both paths are open.

How Did They Prove It? (The Construction)

The authors didn't just guess this; they built a mathematical bridge to prove it exists. Here is their method, simplified:

1. The "Supersolution" (The Safety Net):
   They constructed a "ceiling" or a "safety net" (called a supersolution). Imagine a giant, invisible dome covering the "frozen volcano." This dome is designed to be "hotter" than the volcano but follows the rules of the heat equation. It acts as a container that prevents the heat from exploding out of control.

2. The "Self-Similar" Solution (The Shapeshifter):
   They used a special type of solution called a "self-similar solution." Think of this as a shape that changes size but keeps its proportions (like a fractal). They took this shape and morphed it to fit inside their safety net.

3. The Comparison Principle (The Squeeze):
   They started with a series of "approximate" starting points that were slightly less extreme than the frozen volcano. They let the heat equation run on these approximations.

   • Because of the "safety net" (the supersolution), these approximations couldn't run away to infinity.
   • Because they started below the frozen volcano, they couldn't jump over it.
   • As they made the approximations closer and closer to the real "frozen volcano," the resulting heat waves settled into a new, stable pattern.

4. The Result:
   This new pattern is the "thawing" solution. It starts at the same point as the frozen volcano but immediately begins to move and change. Since the "frozen" version is also a valid solution, uniqueness is broken.

Why Does This Matter?

You might ask, "Who cares if a volcano stays frozen or starts moving in a math problem?"

1. Predictability: It tells us that for certain extreme physical systems, knowing the initial state perfectly is not enough to predict the future. The system has a "choice."
2. The "Joseph-Lundgren" Threshold: The paper identifies a specific "Goldilocks zone" for how fast the heat can grow. If it grows too slow, the system is unique. If it grows too fast, things blow up instantly. But in this specific middle zone, the system is ambiguous.
3. New Tools: The authors developed a new way to construct these solutions without relying on the usual "scale-invariance" tricks (which don't work for this specific type of equation). This opens the door to solving other messy, real-world problems where the rules aren't perfectly symmetrical.

Summary in One Sentence

This paper proves that for a specific type of extreme heat equation, if you start with a perfectly balanced, infinitely hot "frozen explosion," the universe allows two equally valid futures: one where it stays frozen forever, and another where it immediately begins to cool down and spread out.

Technical Summary

Here is a detailed technical summary of the paper "Non-Uniqueness of Positive Solutions for Supercritical Semilinear Heat Equations Without Scale Invariance" by Kotaro Hisa and Yasuhito Miyamoto.

1. Problem Statement

The paper investigates the Cauchy problem for semilinear heat equations with a general nonlinearity $f(u)$ in $\mathbb{R}^N$ ($N > 2$):
$$\begin{cases} \partial_t u - \Delta u = f(u), & x \in \mathbb{R}^N, \, t > 0, \\ u(x, 0) = u_0(x), & x \in \mathbb{R}^N. \end{cases}$$
The primary objective is to establish the non-uniqueness of positive solutions when the initial data $u_0$ is a positive radial singular stationary solution ($u_0 = u_*$).

Specifically, the authors consider a wide class of nonlinearities $f(u)$ that are not necessarily scale-invariant (i.e., not just pure powers $u^p$). The study focuses on the supercritical regime, where the growth rate of $f$ exceeds the critical Sobolev exponent $p_S = \frac{N+2}{N-2}$ but remains below the Joseph-Lundgren exponent $p_{JL}$ (for $N > 10$).

2. Mathematical Framework and Assumptions

The authors impose the following conditions on the nonlinearity $f$:

• Regularity: $f \in C^1[0, \infty) \cap C^2(0, \infty)$, $f(0)=f'(0)=0$, and $f$ is convex ($f'' > 0$).
• Growth Rate: The limit $q_f := \lim_{u \to \infty} \frac{f'(u)^2}{f(u)f''(u)}$ exists. This defines a conjugate growth rate $p_f$ such that $\frac{1}{p_f} + \frac{1}{q_f} = 1$.
• Super-criticality: The growth is supercritical, meaning $p_S < p_f < p_{JL}$ (or equivalently $q_{JL} < q_f < q_S$).
• Existence of Singular Solution: It is assumed that a positive radial singular stationary solution $u_*$ exists, satisfying $-\Delta u_* = f(u_*)$ in $\mathbb{R}^N \setminus \{0\}$ with $u_*(x) \to \infty$ as $|x| \to 0$.

Key Exponents:

• Sobolev Critical: $p_S = \frac{N+2}{N-2}$.
• Joseph-Lundgren: $p_{JL} = 1 + \frac{4}{N-4-2\sqrt{N-1}}$ (for $N > 10$).
• Conjugate Exponents: $q_S = \frac{p_S}{p_S-1}$ and $q_{JL} = \frac{p_{JL}}{p_{JL}-1}$.

3. Methodology

The proof of non-uniqueness relies on constructing two distinct solutions starting from the same singular initial data $u_*$:

1. The Singular Stationary Solution: $u(x,t) = u_*(x)$ for all $t > 0$.
2. A Bounded Solution: A solution $u(x,t)$ that becomes bounded for $t > 0$ and converges to $u_*$ as $t \to 0^+$.

Construction of the Bounded Solution:
The core of the methodology involves a monotonicity argument combined with a comparison principle, rather than the contraction mapping theorem.

• Step 1: Regularization. Approximate the singular initial data $u_*$ by bounded functions $u_{0,n} = \min\{u_*, n\}$. Since $u_{0,n}$ is bounded and uniformly continuous, the problem admits a unique classical solution $u_n(t)$.
• Step 2: Construction of a Supersolution. The authors construct a global supersolution $v(x,t)$ that bounds the sequence $u_n(t)$ from above. This $v$ is formed by "gluing" two components:
  • The singular solution $u_*$ in the outer region ($|x| \geq r(t)$).
  • A transformed forward self-similar solution in the inner region ($|x| < r(t)$).
• Step 3: Self-Similar Transformation. For nonlinearities like $u^p$ or $e^u$, the authors utilize forward self-similar solutions of the form $U(x,t) = \phi(|x|/\sqrt{t})$. They define a transformation $\bar{u}$ based on these self-similar profiles. Crucially, they prove that for a specific choice of parameters, this transformed function acts as a supersolution near the origin where the singularity is "active."
• Step 4: Intersection Analysis. Using the properties of the intersection number between regular and singular solutions of the associated ODEs, they show that the transformed self-similar solution and the singular stationary solution intersect at a moving radius $r(t)$ which tends to 0 as $t \to 0$.
• Step 5: Limit Process. The sequence $u_n(t)$ is monotone increasing and bounded by $v(t)$. By the Monotone Convergence Theorem, the limit $u(t) = \lim_{n \to \infty} u_n(t)$ exists. This limit satisfies the integral formulation of the heat equation and is a bounded solution distinct from $u_*$.

4. Key Results

Theorem 1.4 (Supercritical Non-Uniqueness):
Under assumptions (A1)–(A6), if $u_*$ is a positive radial singular stationary solution, then the Cauchy problem with $u_0 = u_*$ has at least two positive solutions:

1. The stationary solution $u(x,t) = u_*(x)$.
2. A solution $u(x,t)$ such that $u \in L^\infty_{loc}((0, t_0); L^\infty(\mathbb{R}^N))$ and $u(t) \to u_*$ in $L^\gamma_{ul}(\mathbb{R}^N)$ as $t \to 0^+$ for $1 \leq \gamma < \gamma^*$.

Theorem 1.5 (Critical/Subcritical Non-Uniqueness):
The result is extended to the critical and subcritical cases ($p_0 < p_f \leq p_S$) provided a singular solution satisfying specific asymptotic behavior near the origin exists.

Reduction to Existence:
A major contribution is the reduction of the non-uniqueness problem to the existence problem of a positive radial singular stationary solution. If such a solution exists (which is guaranteed in the supercritical range $q_{JL} < q_f < q_S$), non-uniqueness follows immediately.

5. Examples and Applications

The paper provides three specific examples of nonlinearities satisfying the conditions:

1. Exponential-Power Growth: $f(u) = u^\beta e^{u^\gamma}$ with $\beta \geq p_S, \gamma > 1$. This covers the case where $q_f = 1$ (exponential type), valid for $2 < N < 10$.
2. Piecewise Defined Exponential: A constructed function behaving like $u^5 e^{au}$ near zero and $e^{au}$ for large $u$.
3. Sum of Powers: $f(u) = u^\beta + u^\gamma$ with $p_S < \gamma < \beta < p_{JL}$.

6. Significance and Contributions

• Generalization of Scale-Invariance: Previous non-uniqueness results largely relied on the scale-invariance of pure power nonlinearities ($u^p$). This paper extends the theory to general nonlinearities without scale invariance, including exponential and mixed growth types.
• Unified Approach: The authors unify the treatment of power-type and exponential-type nonlinearities using the conjugate exponent $q_f$ and the concept of "quasi-scaling" via transformations of self-similar solutions.
• Methodological Innovation: The construction of the supersolution by combining a singular stationary solution with a transformed self-similar solution is a novel technique. It avoids the direct use of "proper solutions" (a concept from Galaktionov-Vazquez) and instead works directly within the framework of Definition 1.1 (integral solutions), making the result applicable to a broader class of function spaces.
• Sharpness of Conditions: The results clarify the role of the Joseph-Lundgren exponent. Non-uniqueness holds in the range where singular solutions exist and are stable in a specific sense, bridging the gap between the critical Sobolev exponent and the Joseph-Lundgren exponent.

In summary, the paper establishes that the existence of a singular stationary solution is sufficient to trigger non-uniqueness for a very wide class of supercritical semilinear heat equations, regardless of whether the nonlinearity possesses exact scale invariance.