Nonlinear parabolic problem with time fractional derivative

1. Problem Statement

The authors investigate a time-fractional evolutionary partial differential equation (PDE) involving the $p$-Laplacian operator and a double singular Hardy-type potential. The specific problem is defined on a bounded, star-shaped domain $\Omega \subset \mathbb{R}^n$ ($n > p > 2$) with respect to a small ball centered at the origin.

The governing equation is:
$$\begin{cases} D^\alpha_t u - \Delta_p u - \mu W(x)|u|^{p-2}u = 0, & \text{in } Q = (0, T) \times \Omega, \\ u(t, x) = 0, & \text{on } [0, T] \times \partial\Omega, \\ u(0, x) = u_0(x), & \text{in } \Omega, \end{cases}$$
where:

• $D^\alpha_t$ is the Liouville-Caputo fractional derivative of order $\alpha \in (0, 1)$.
• $\Delta_p u = \text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian.
• $\mu > 0$ is a parameter.
• The potential $W(x)$ is singular both at the origin and on the boundary $\partial\Omega$:
  $$W(x) = |x|^{-p} \left[ 1 - \left( \frac{|x|}{\phi(x)} \right)^{\frac{n-p}{p}} \right]^{-p}$$
  where $\phi(x)$ defines the boundary geometry.

The central question is the global existence versus finite-time blow-up of weak solutions, depending on the relationship between the parameter $\mu$ and the optimal Hardy constant $\Lambda(n, p) = \left(\frac{n-p}{p}\right)^p$.

2. Methodology

The authors employ a multi-step analytical approach combining fractional calculus, nonlinear functional analysis, and comparison principles.

A. Fractional Ordinary Differential Equations (FODE)

Before tackling the PDE, the authors establish foundational results for FODEs:

• Comparison Principle: They prove a comparison principle for the equation $D^\alpha_t w = f(t, w)$ under Lipschitz conditions on $f$.
• Blow-up Analysis: They analyze the specific FODE $D^\alpha_t u = u^p$ ($p>1$). By constructing a supersolution and utilizing integral representations, they prove that solutions to this scalar equation blow up in finite time $T_m < \infty$. This result is crucial for establishing the blow-up mechanism in the PDE context.

B. Comparison Principle for the PDE

To handle the nonlinear $p$-Laplacian and fractional time derivative, the authors derive a comparison principle for the PDE:

• They utilize a "fundamental identity" (adapted from Zacher) involving the fractional derivative of a convex function (specifically $H(z)=z^2$).
• By testing the difference of two solutions against the positive part of their difference, they prove that if $u(0) \leq v(0)$, then $u(t) \leq v(t)$ for all $t$, provided the potential term is bounded.

C. Approximation and A Priori Estimates

Since the potential $W(x)$ is singular, the authors introduce a truncated problem where $W(x)$ is replaced by $W_N(x) = \min\{N, W(x)\}$.

• A Priori Estimates: For the truncated problem, they derive energy estimates by testing the equation with $u_N$ and integrating over the domain.
• They leverage the Hardy Inequality:
  $$\int_\Omega |\nabla u|^p dx \geq \Lambda(n, p) \int_\Omega W(x)|u|^p dx$$
  This inequality is pivotal in controlling the singular potential term when $\mu < \Lambda(n, p)$.

D. Limit Process

The solution to the original problem is defined as the limit of the solutions to the truncated problems as $N \to \infty$. The existence of this limit relies on the uniform bounds derived from the a priori estimates.

3. Key Contributions and Results

The paper establishes two main theorems characterizing the behavior of weak solutions based on the parameter $\mu$:

Theorem 1.1: Global Existence (Sub-critical Case)

• Condition: $\mu < \Lambda(n, p)$.
• Result: If the initial data $u_0 \in W^{1,p}_0(\Omega)$, then every weak solution exists globally in time (i.e., on $(0, \infty)$).
• Mechanism: The sub-critical potential is dominated by the diffusion term ($p$-Laplacian). The a priori estimates ensure that the $L^2$ and $W^{1,p}$ norms remain bounded, preventing blow-up.

Theorem 1.2: Finite-Time Blow-up (Super-critical Case)

• Condition: $\mu > \Lambda(n, p)$ and $u_0(x) > 0$.
• Result: No global weak solution exists; the solution blows up in finite time.
• Mechanism: The super-critical potential overwhelms the diffusion. The authors construct a positive subsolution of the form $v(t, x) = T(t)X(x)$, where $X(x)$ is an eigenfunction of the truncated operator and $T(t)$ is a solution to the blow-up FODE ($D^\alpha_t T = T^{p-1}$). By the comparison principle, the actual solution $u$ is bounded below by this blowing-up subsolution, forcing $u$ to blow up at the same finite time.

4. Significance and Impact

1. Extension to Fractional Time: This work extends classical results on singular parabolic problems (previously studied for integer-order time derivatives) to the time-fractional regime. This is significant for modeling physical processes with memory effects (e.g., anomalous diffusion in porous media).
2. Double Singularity: The paper addresses a potential singular at both the origin and the boundary, a more complex scenario than standard inverse-square potentials.
3. Sharp Threshold: The authors identify the optimal Hardy constant $\Lambda(n, p)$ as the precise threshold separating global existence from finite-time blow-up. This provides a complete classification of the solution behavior based on the strength of the singular potential.
4. Methodological Rigor: The combination of the fundamental identity for fractional derivatives with the comparison principle for $p$-Laplacian equations offers a robust framework for analyzing similar nonlinear fractional evolutionary problems.

In summary, the paper provides a rigorous mathematical framework for understanding how memory effects (fractional time) and strong singularities interact in nonlinear diffusion processes, determining exactly when the system remains stable and when it collapses.