Lipschitz regularity for parabolic fractional $p$-Laplace equations

1. Problem Statement

The paper addresses the regularity of local weak solutions to nonlocal parabolic $p$-Laplace equations of the form:
$$\partial_t u + L_K u = 0 \quad \text{in } \Omega \times I$$
where the nonlocal operator $L_K$ is defined as:
$$L_K u(x, t) := \text{P.V.} \int_{\mathbb{R}^N} |u(x, t) - u(y, t)|^{p-2}(u(x, t) - u(y, t)) K(x-y) \, dy$$
Key Constraints and Assumptions:

• Parameters: $1 < p < \infty$ and $s \in (0, 1)$.
• Kernel: $K$ is symmetric, translation-invariant, and satisfies standard ellipticity bounds: $\frac{\lambda}{|z|^{N+sp}} \leq K(z) \leq \frac{\Lambda}{|z|^{N+sp}}$. Crucially, $K$ is not assumed to be continuous.
• Regime: The study focuses on the condition $sp > p - 1$.
• Goal: To prove that solutions are locally Lipschitz continuous in space, uniformly in time.

2. Methodology

The proof employs a sophisticated combination of scaling arguments, truncation techniques, and the Ishii–Lions method (originally developed for viscosity solutions of local PDEs), adapted here for nonlocal nonlinear operators.

A. Scaling and Normalization

The authors reduce the general problem to a normalized setting on a fixed cylinder $B_2 \times (-2, 0]$.

• They define a normalization constant $M_R$ involving the $L^\infty$ norm and the "tail" of the solution.
• Dichotomy in Scaling: The scaling strategy differs based on $p$:
  • For $p \geq 2$, large amplitudes enlarge the time scale.
  • For $1 < p < 2$, large amplitudes shrink the spatial scale.
• This reduction allows the authors to work with a solution $u$ where $\|u\|_{L^\infty} + \text{Tail} \leq 1$.

B. The Truncation Strategy (The "Touching Lemma")

A major technical hurdle in nonlocal equations is the dependence on the solution's behavior at infinity (the "tail").

• The operator is decomposed: $L_K u = L_{B_1}^K u + L_{B_1^c}^K u$.
• The "tail" part $L_{B_1^c}^K u$ is treated as a bounded source term $f(x,t)$ (due to the tail assumption $u \in L^\infty_{\text{loc}}(I; L^{p-1}_{sp})$).
• The main equation becomes $\partial_t u + L_{B_1}^K u = f$.
• Key Advantage: The truncated operator $L_{B_1}^K$ only integrates over a bounded ball. This removes the need for the test function to have a bounded tail, allowing the use of smooth test functions with compact support without continuity assumptions on the kernel $K$.

C. The Ishii–Lions Framework

To prove Lipschitz continuity, the authors use a contradiction argument based on the Ishii–Lions method:

1. Penalization: They define a gap function $g(x, y, t) = u(x, t) - u(y, t) - \phi_L(|x-y|) - m_1 \psi(x)$, where $\phi_L$ is a penalization profile.
2. Doubling of Variables: They consider a time-doubled functional $\Phi_\nu$ to find a maximum point $(x_\nu, y_\nu, t_\nu, \tau_\nu)$.
3. Touching Lemma: At the maximum points, they construct smooth test functions $w_1$ and $w_2$ that touch $u$ from above and below. Since $\nabla u \neq 0$ at these points (ensured by the penalization), the Touching Lemma (Proposition 3.8) applies, yielding inequalities for the time derivatives and the truncated operators.
4. Master Inequality: Subtracting the inequalities for $w_1$ and $w_2$ eliminates the time derivative term, resulting in a master inequality involving the difference of the nonlocal operators:
   $$L_{B_1}^K w_1(x_\nu, t_\nu) - L_{B_1}^K w_2(y_\nu, \tau_\nu) \leq 2C_0$$
5. Integral Estimates: The left-hand side is decomposed into four geometric regions (a cone near the diagonal, an annulus, and the outer region).
   • Hölder Bootstrap: First, they prove higher Hölder regularity iteratively to reach an exponent $\kappa$ close to 1.
   • Lipschitz Step: They use a specific almost-linear profile $\tilde{\phi}(r) = r + \frac{r}{\log r}$ (strictly concave) to handle the $p$-Laplacian structure. This profile allows them to derive a contradiction as the penalization parameter $L \to \infty$, provided $sp > p-1$.

3. Key Contributions

1. Full Range of $p$: The result holds for all $1 < p < \infty$, including the subquadratic case ($1 < p < 2$). Previous works often struggled with the subquadratic case due to singularities in the operator evaluation on smooth functions; this paper circumvents this via the specific choice of test functions and the truncation strategy.
2. Discontinuous Kernels: The proof works for kernels $K$ that are not continuous, provided they satisfy the ellipticity bounds. This generalizes previous results that required explicit kernels like $K(z) = |z|^{-(N+sp)}$.
3. Weaker Tail Assumption: The paper requires only that the tail of the solution is bounded in time ($u \in L^\infty_{\text{loc}}(I; L^{p-1}_{sp})$), whereas recent independent works (e.g., Jesus–Sobral–Urbano) required continuity of the tail in time ($u \in C_{\text{loc}}(I; L^{p-1}_{sp})$).
4. Avoidance of Blow-up/Liouville: Unlike linear parabolic regularity proofs that rely on heat kernel estimates, blow-up arguments, or Liouville theorems, this proof is purely local and structural, relying on the Ishii–Lions framework.

4. Main Results

Theorem 1.1 (Spatial Lipschitz Regularity):
Let $u$ be a local weak solution to the parabolic fractional $p$-Laplace equation with $sp > p-1$. Then $u(\cdot, t)$ is locally Lipschitz continuous in space, uniformly in time. Specifically, for a cylinder $Q_{2R, 2R^{sp}}$:
$$\sup_{t} [u(\cdot, t)]_{C^{0,1}(B_{R/2})} \leq C M_R R^{-1} \quad \text{if } p \geq 2$$
$$\sup_{t} [u(\cdot, t)]_{C^{0,1}(B_{R/2})} \leq C M_R^{1 + \frac{2-p}{sp}} R^{-1} \quad \text{if } 1 < p < 2$$
where $M_R$ depends on the $L^\infty$ norm and the tail of $u$.

Sharpness: The condition $sp > p-1$ is shown to be sharp for this method (based on counterexamples in the elliptic case).

5. Significance

• Bridging Elliptic and Parabolic: This work extends the breakthrough by Biswas–Topp (2025) on elliptic fractional $p$-Laplace equations to the parabolic setting, establishing a unified regularity theory.
• Robustness: By handling discontinuous kernels and weaker tail assumptions, the results are applicable to a much broader class of physical and probabilistic models where the kernel may be rough or the solution's long-range behavior is only bounded.
• Methodological Innovation: The successful adaptation of the Ishii–Lions method to the nonlocal, nonlinear, parabolic setting without relying on viscosity solution theory (which was recently established for $p>2$ by others) provides a new, direct route to regularity. The use of the "almost-linear" penalization profile is a critical technical innovation to handle the logarithmic corrections needed for the Lipschitz limit.
• Extensions: The methodology is shown to extend to fractional $(p,q)$-Laplacians and nonlocal double-phase operators, suggesting a versatile framework for future regularity studies in nonlocal PDEs.