Sharp non-explicit blow-up profile for perturbed nonlinear heat equations with gradient terms

1. Problem Statement

The paper investigates the blow-up phenomenon for a class of perturbed nonlinear heat equations involving gradient terms in $\mathbb{R}^N$:
$$\begin{cases} u_t = \Delta u + |u|^{p-1}u + h(u, \nabla u), \\ u(x, 0) = u_0 \in W^{1,\infty}(\mathbb{R}^N), \end{cases}$$
where $1 < p$, and the perturbation term $h$ satisfies specific growth conditions: $|h(u, v)| \le C(|u|^\gamma + |v|^\gamma + 1)$ with $0 \le \gamma < p$ and $0 \le \gamma < \frac{2p}{p+1}$.

The unperturbed case ($h \equiv 0$) is well-studied, with known results regarding single-point blow-up and the asymptotic profile (specifically the work of Ebde and Zaag, [EZ11]). However, two critical questions remained open for the perturbed case:

1. Single-point blow-up: Is the blow-up point $a$ unique, or can blow-up occur at multiple points?
2. Final blow-up profile: Can one define a precise final profile $u^*(x) = \lim_{t \to T} u(x,t)$ for $x \neq a$, and determine its behavior near the singularity?
3. Sharpness of estimates: Can the error estimates in the blow-up profile be improved from the slow logarithmic scale ($1/|\log(T-t)|$) to a sharper scale involving the natural time variable $(T-t)$?

2. Methodology

The author employs a rigorous analysis based on self-similar variables and spectral decomposition techniques.

A. Self-Similar Variables

The equation is transformed using the variables:
$$y = \frac{x-a}{\sqrt{T-t}}, \quad s = -\log(T-t), \quad w(y,s) = (T-t)^{\frac{1}{p-1}}u(x,t).$$
This transforms the PDE into a parabolic equation for $w$:
$$w_s = \Delta w - \frac{1}{2}y \cdot \nabla w - \frac{w}{p-1} + |w|^{p-1}w + e^{-\frac{p}{p-1}s}h(\dots).$$
The solution is decomposed around a known profile $f(y/\sqrt{s})$ (from the unperturbed case) plus a remainder term $q(y,s)$.

B. The Shrinking Set $V_A(s)$

The proof relies on defining a "shrinking set" $V_A(s)$ of functions that control the remainder $q$. This set ensures that the solution stays close to the profile $f$ as $s \to \infty$. The set is defined by bounds on the spectral components of $q$ (projections onto Hermite polynomials) and the tail of the solution.

C. Comparison of Two Solutions

A central technique is comparing two distinct blow-up solutions, $u_1$ and $u_2$ (potentially with different perturbations $h_1, h_2$ or different blow-up times/points). The difference $g = w_1 - w_2$ is analyzed.

• Spectral Analysis: The difference $g$ is expanded in the eigenbasis of the linearized operator $L = \Delta - \frac{y}{2}\cdot\nabla + 1$. The evolution of the coefficients (modes) is studied to show that the difference decays rapidly.
• Parabolic Regularity: New estimates are derived to control the gradient term $\nabla u$ in the perturbation $h$, which is the main difficulty compared to the unperturbed case. This involves iterative arguments and Gronwall-type inequalities.
• No-Blow-Up Threshold: A crucial step involves proving that if the solution is sufficiently small in a specific region (near the blow-up point but outside the core singularity), it cannot blow up. This is achieved by rescaling the solution and using Duhamel's formula combined with heat semigroup smoothing properties.

3. Key Contributions and Results

A. Single-Point Blow-Up (Theorem 1)

The paper proves that for any solution $u$ in the constructed set $S_{a,T,h}$ (solutions that blow up at $T$ and $a$ with specific asymptotic behavior), $a$ is the unique blow-up point.

• Result: The solution blows up only at $x=a$.
• Final Profile: A final profile $u^*(x)$ exists for all $x \neq a$.
• Asymptotic Behavior: Near the blow-up point, the final profile behaves as:
  $$u^*(x) \sim \left( \frac{8p |\log|x-a||}{(p-1)^2 |x-a|^2} \right)^{\frac{1}{p-1}}$$
  with a sharp error estimate involving logarithmic corrections.

B. Sharp Non-Explicit Blow-Up Profile (Theorem 2)

This is the paper's most significant technical contribution. Previous results (e.g., [EZ11]) described the blow-up profile using an explicit function $f$ with an error of order $O(|\log(T-t)|^{-1})$.

• Improvement: The author shows that the solution $u(x,t)$ is actually much closer to a non-explicit solution $\hat{u}$ of the unperturbed equation (where $h \equiv 0$).
• The Estimate:
  $$|u(x,t) - \hat{u}(x-a, t-T+1)| \le C \max \left( (T-t)^{-\frac{1}{p-1}}|\log(T-t)|, |x-a|^{-\frac{2}{p-1}}|\log|x-a||^{\frac{1}{4} - \frac{1}{p-1}} \right).$$
• Significance: By replacing the explicit profile $f$ with the solution $\hat{u}$ of the unperturbed equation, the error term is significantly reduced. The error is now controlled in the "fast" variable scale $(T-t)$ rather than just the slow logarithmic scale.
• 1D Refinement: In one dimension ($N=1$), the author derives an even sharper expansion involving a scaling parameter $\lambda$, providing a Taylor-like expansion of the difference between the perturbed and unperturbed solutions.

4. Technical Challenges Overcome

• Gradient Term Control: The presence of $h(u, \nabla u)$ introduces nonlinearity in the gradient. The author had to develop new parabolic regularity estimates to bound $\nabla w$ and show that the gradient term does not destabilize the blow-up profile.
• Non-Explicit Profile: Proving that the solution converges to a non-explicit profile (a solution of the unperturbed equation) rather than an explicit function requires a delicate comparison argument between two solutions, rather than just linearizing around a fixed function.
• Spectral Gap: Managing the interaction between the unstable, neutral, and stable modes of the linearized operator in the presence of the perturbation $h$.

5. Significance

• Refinement of Theory: This work extends the classical theory of blow-up profiles (developed for $h \equiv 0$ by Merle, Zaag, Velázquez, etc.) to equations with gradient perturbations.
• Optimality: It demonstrates that the "natural" blow-up profile for perturbed equations is not a simple explicit function but is structurally identical to the unperturbed solution, just shifted and scaled. This suggests a form of structural stability of the blow-up profile under gradient perturbations.
• Methodological Advance: The techniques used to control the gradient term and establish the "no blow-up under threshold" property provide a robust framework for analyzing other perturbed semilinear parabolic equations.

In summary, Boughrara establishes that despite the presence of gradient perturbations, the blow-up dynamics remain governed by the unperturbed heat equation's structure, allowing for a much sharper description of the singularity than previously possible.