Global existence and blow-up for the Hardy-Sobolev parabolic equation in RN

Technical Summary: Global Existence and Blow-up for the Hardy-Sobolev Parabolic Equation in $\mathbb{R}^N$

Problem Statement
This paper investigates the long-time behavior and finite-time blow-up properties of solutions to the following classical semilinear parabolic equation involving a critical Sobolev exponent and a Hardy potential in $\mathbb{R}^N$ ($N \ge 3$):
$$\begin{cases} u_t - \Delta u - \frac{\mu u}{|x|^2} = |u|^{p-1}u & \text{in } \mathbb{R}^N \times (0, T), \\ u(x, 0) = u_0(x) & \text{in } \mathbb{R}^N, \end{cases}$$
where $\mu \in [0, (N-2)^2/8]$ and $p = 2^* - 1$ with $2^* = 2N/(N-2)$. The equation models phenomena in quantum mechanics, chemistry, and cosmology, particularly where inverse square potentials arise. The study focuses on the dichotomy between global existence and finite-time blow-up for initial data with low, critical, and high energy levels.

Methodology
The authors employ a two-pronged analytical approach:

1. Self-Similar Transformation: The original equation is transformed via the change of variables $v(y, s) = (1+t)^{1/(p-1)}u(x, t)$, $t = e^s - 1$, and $x = (1+t)^{1/2}y$. This converts the original problem into a parabolic equation with a linear drift term and a source term:
   $$v_s - \Delta v - \frac{1}{2}y \cdot \nabla v = \beta v + \frac{\mu v}{|y|^2} + |v|^{p-1}v,$$
   where $\beta = (N-2)/4$. This transformation allows the analysis of asymptotic behavior through the lens of a weighted operator.
2. Modified Potential Well Method: The authors utilize a modified potential well framework, originally developed by Payne and Sattinger and later refined by Liu et al. This involves defining energy functionals ($E_K$), Nehari manifolds ($N$), and potential wells ($W, V$) based on the sign of the functional $D_K(v) = A(v) - B(v)$.
   • A weighted Hardy inequality is established to handle the singular potential term $\mu/|y|^2$ within the weighted Hilbert space $X$ equipped with the weight $K(y) = e^{|y|^2/4}$.
   • The analysis distinguishes between three energy regimes: sub-critical ($E_K(v_0) < d$), critical ($E_K(v_0) = d$), and super-critical ($E_K(v_0) > d$), where $d$ represents the depth of the potential well.

Key Contributions and Results

• Establishment of a Weighted Hardy Inequality: The paper proves a specific weighted Hardy inequality (Lemma 2.2) with the optimal constant $C_N = (\frac{2}{N-2})^2$ in the weighted space $L^2_K(\mathbb{R}^N)$. This is crucial for controlling the singular term in the energy estimates.
• Low Initial Energy ($E_K(v_0) < d$):
  • Global Existence: If the initial data satisfies $D_K(v_0) > 0$, the solution exists globally in time and remains within the potential well $W$.
  • Blow-up: If $D_K(v_0) < 0$, the solution blows up in finite time.
  • Decay Rate: For global solutions with low energy, the authors derive a precise exponential decay rate for the $L^2_K$ norm: $\|v(s)\|_{K,2}^2 \le \|v_0\|_{K,2}^2 e^{-\frac{1}{2}(1-\delta_1)s}$, where $\delta_1$ is a root of the equation $d(\delta) = E_K(v_0)$.
• Critical Initial Energy ($E_K(v_0) = d$):
  • Global Existence: If $D_K(v_0) \ge 0$, global weak solutions exist.
  • Blow-up: If $D_K(v_0) < 0$, solutions blow up in finite time.
  • The paper extends the potential well method to the critical energy level, showing that solutions starting on the boundary of the well behave similarly to those inside, provided the Nehari functional is non-negative.
• High Initial Energy ($E_K(v_0) > d$):
  • The authors provide sufficient conditions for global existence and blow-up even when the initial energy exceeds the potential well depth.
  • Specifically, if $v_0 \in N_+$ and $\|v_0\|_{K,2} \le \lambda_{E_K(v_0)}$, the solution exists globally and converges to zero. Conversely, if $v_0 \in N_-$ and $\|v_0\|_{K,2} \ge \Lambda_{E_K(v_0)}$, the solution blows up in finite time.
• Asymptotic Behavior: For global solutions that are uniformly bounded, the $\omega$-limit set contains a stationary solution. The paper demonstrates that the global solution behaves asymptotically like a Palais-Smale sequence for the associated stationary equation, converging weakly to a non-trivial stationary solution or zero.

Significance and Claims
The paper claims to generalize existing results on semilinear parabolic equations by incorporating the Hardy potential and the critical Sobolev exponent simultaneously.

• Generality: The results for low initial energy are presented as more general than previous works (e.g., [7]), specifically by proving a more precise decay rate for the $L^2$ norm of global solutions.
• Methodological Extension: The work successfully adapts the modified potential well method to the context of the Hardy-Sobolev parabolic equation, handling the singular potential and the critical growth term through the self-similar transformation and weighted functional analysis.
• Completeness: The study provides a comprehensive classification of solution behavior across all energy levels (low, critical, and high), establishing clear thresholds for global existence versus finite-time blow-up based on the initial data's position relative to the Nehari manifold and energy depth.

The authors do not propose new experimental applications but position their work as a rigorous theoretical advancement in the analysis of evolution equations with singular potentials, contributing to the understanding of the competition between diffusion, linear drift, and nonlinear growth in the presence of inverse-square potentials.