Local existence and blow-up behavior for the diffusive Hamilton-Jacobi equation in a half-space with unbounded initial data

Technical Summary: Local Existence and Blow-up Behavior for the Diffusive Hamilton-Jacobi Equation in a Half-Space with Unbounded Initial Data

Problem Statement
This paper investigates the diffusive Hamilton-Jacobi equation:
$$u_t - \Delta u = |\nabla u|^p, \quad p > 1$$
posed in the half-space $H = \mathbb{R}^n_+$ with Dirichlet boundary conditions ($u=0$ on $\partial H$). The primary focus is on solutions with unbounded initial data $u_0(x)$ that may grow at spatial infinity. While the behavior of solutions in bounded domains and for bounded initial data in unbounded domains is well-established (notably the phenomenon of gradient blow-up, or GBU), the case of unbounded initial data in unbounded domains remains largely unexplored. The paper addresses the local well-posedness, the sharpness of admissible growth rates, and the detailed blow-up dynamics for such solutions.

Methodology
The analysis relies on a combination of comparison principles, barrier function constructions, rescaling arguments, and Liouville-type theorems.

1. Local Well-Posedness and Uniqueness: The authors establish a comparison principle (Proposition 3.1) tailored for solutions with polynomial growth at infinity. This involves a penalization argument at spatial infinity using a perturbed function $Z_\epsilon$ to handle the unbounded domain. Uniqueness is derived as a consequence of this comparison principle within a specific function space $Y_T$.
2. Existence via Regularization: Local existence is proven by regularizing the initial data $u_0$ via mollification to obtain bounded, smooth approximations $u_{0,k}$. Uniform a priori estimates (specifically on the solution and its gradient) are derived using supersolutions and barrier functions (e.g., $v_1$ and $v_2$) that depend on the growth rate of $u_0$. A compactness argument then yields a classical solution.
3. Non-existence: To prove the sharpness of the admissible growth class, the authors construct a contradiction by comparing a hypothetical solution with a backward self-similar solution from previous literature [5]. If the initial data grows too rapidly, it dominates the self-similar profile, leading to a contradiction with the existence of a classical solution up to a certain time.
4. Blow-up Analysis:
   • Type I vs. Type II: The paper distinguishes between Type I blow-up (where the gradient grows at the self-similar rate) and Type II blow-up (faster growth).
   • Rescaling and Liouville Theorems: For Type II blow-up, the authors employ a rescaling argument (blow-up analysis) where the solution is zoomed in near the singularity. The limit of these rescaled solutions is shown to be an entire solution of the equation. The classification of these entire solutions relies on recent Liouville-type theorems [5, 8], which assert that bounded entire solutions in the half-space must be stationary and depend only on the normal variable $x_n$.
   • Gradient Estimates: Bernstein-type estimates and heat semigroup regularity arguments are used to derive precise upper and lower bounds on the gradient near the blow-up time.

Key Contributions and Results

1. Optimal Class of Initial Data:
   The paper defines an optimal class of initial data $X$ (for $1 < p \le 2$) and $X_1$ (for $p > 2$) allowing for growth at infinity and singularities at the boundary.

   • Growth at Infinity: The critical exponent is identified as $p/(p-1)$. Specifically, $u_0(x) \lesssim x_n^{1+\beta}$ as $x_n \to \infty$ (where $\beta = 1/(p-1)$) is the sharp threshold. Theorem 2.3 proves that if $u_0$ grows faster than this rate uniformly in $x'$, no classical solution exists.
   • Boundary Behavior: For $p > 2$, the initial data must satisfy a specific bound near the boundary $x_n=0$: $\limsup_{x_n \to 0} x_n^{\beta-1} \sup_{x'} u_0(x', x_n) < c_p$. If this is violated, the solution fails to be classical immediately.

2. Local Well-Posedness and Blow-up Alternative:

   • Theorems 2.1 & 2.2: For initial data in the optimal class, there exists a unique maximal classical solution.
   • Blow-up Alternative: If the maximal existence time $T_{max}$ is finite, the solution either blows up in the weighted $L^\infty$ norm (for $1 < p \le 2$) or exhibits gradient blow-up (GBU) at a finite distance from the boundary (for $p > 2$).

3. Blow-up Behavior for Subcritical Growth:
   When the initial data has subcritical growth ($u_0 \lesssim x_n^\gamma$ with $\gamma < 1+\beta$):

   • Global Existence: For $1 < p \le 2$, solutions exist globally in time.
   • Finite-Time Blow-up: For $p > 2$, finite-time blow-up occurs only via Type II gradient blow-up on the boundary. The solution remains bounded on any strip $\Gamma_R$, while the gradient becomes unbounded.
   • Sharp Estimates: The paper derives sharp upper and lower bounds for the gradient. The lower bound matches the standard Type II rate: $\|\nabla u\|_{L^\infty} \ge C(T-t)^{-1/(p-2)}$. The upper bound shows the gradient is controlled by the singular stationary profile: $|\nabla u| \le (1+\epsilon) d_p x_n^{-\beta} + C$.

4. Asymptotic Profile of Type II Blow-up:

   • Theorem 2.5 (Bubbling): For Type II solutions, there exists a sequence of points converging to the boundary and blow-up time such that the rescaled gradient converges locally uniformly to an explicit one-dimensional profile: $d_p (x_n + \beta)^{-\beta} e_n$.
   • Theorem 2.6 (Dichotomy): At any point near the singularity, the gradient either behaves in a Type I manner or is controlled by the singular stationary profile. This establishes that these are the only two local possibilities for the blow-up behavior.

Significance and Claims
The paper claims to fill a significant gap in the theory of nonlinear parabolic equations by providing the first comprehensive study of the diffusive Hamilton-Jacobi equation in an unbounded domain with unbounded initial data.

• Extension of Well-Posedness: It extends the classical well-posedness framework (typically restricted to bounded data) to an optimal class of unbounded data, identifying the precise critical growth exponent $p/(p-1)$ as the threshold for existence.
• Refined Blow-up Dynamics: Unlike the bounded domain case where blow-up is always Type II boundary GBU, this work reveals a richer structure in the half-space. It demonstrates the coexistence of Type I $L^\infty$ blow-up (for critical growth data) and Type II boundary GBU (for subcritical data).
• Sharpness: The results are shown to be sharp; the paper provides non-existence results for data growing faster than the critical rate and demonstrates that the boundary condition on the initial data for $p>2$ is necessary for classical solvability.
• Asymptotic Description: The paper provides a refined description of the singularity formation, proving that Type II blow-up solutions converge to an explicit one-dimensional profile after rescaling, thereby characterizing the asymptotic singularity formation in detail.

The authors note that their study is restricted to non-negative initial data for simplicity, though they suggest this restriction could be relaxed. The work relies heavily on existing Liouville-type theorems and comparison principles, adapting them to the specific geometry and growth conditions of the half-space problem.