On the Hölder continuity of signed solutions to doubly nonlinear parabolic equations in the mixed degenerate/singular cases

1. Problem Statement

The paper investigates the interior Hölder continuity of sign-changing (signed) weak solutions to a class of doubly nonlinear parabolic equations of the form:
$$\frac{\partial}{\partial t} \left( |u|^{q-1}u \right) - \text{div} \left( A(x, t, u, Du) \right) = 0$$
defined in a domain $\Omega_T = \Omega \times (0, T) \subset \mathbb{R}^N \times \mathbb{R}$.

The vector field $A$ satisfies standard $p$-growth and coercivity conditions:
$$A(x, t, u, \xi) \cdot \xi \geq K_1 |\xi|^p, \quad |A(x, t, u, \xi)| \leq K_2 |\xi|^{p-1}$$
where $K_1, K_2 > 0$.

The Core Challenge:
The equation exhibits both degeneracy (where the diffusion coefficient vanishes) and singularity (where it blows up) depending on the values of $u$ and $Du$. Specifically:

• Degeneracy/Singularity in time: Occurs at $u=0$ depending on $q$.
• Degeneracy/Singularity in space: Occurs at $Du=0$ depending on $p$.

The Gap in Literature:
While Hölder continuity is well-established for:

1. Non-negative solutions.
2. The "doubly singular" case ($p-1 < q$ and $1 < p < 2$).
3. The "doubly degenerate" case ($q < p-1$ and $p > 2$).
4. The borderline case ($q = p-1$).

The specific mixed cases for sign-changing solutions had not been rigorously treated:

• Case A: $0 < q < p-1$ and $p < 2$ (Degenerate in time, Singular in space).
• Case B: $q > p-1$ and $p > 2$ (Singular in time, Degenerate in space).

2. Methodology

The author employs a combination of De Giorgi's iteration technique, DiBenedetto's intrinsic scaling method, and new integral Harnack-type inequalities specifically tailored for sign-changing solutions.

Key Technical Steps:

1. Intrinsic Scaling:
   The proof relies on constructing cylinders with "intrinsic" geometry adapted to the solution's oscillation $\omega$. The time scaling is defined as:
   $$Q_{r, b} \sim B_r \times (t_0 - b \omega^{q-p+1} r^p, t_0)$$
   This scaling balances the time derivative term $|u|^{q-1}u_t$ with the spatial divergence term $|Du|^{p-2}Du$.

2. Integral Harnack Inequalities for Sign-Changing Solutions:
   A major novelty is the derivation of integral Harnack inequalities (Theorems 1.2 and 1.3) for the quantities $\pm(|\mu_\pm|^{q-1}\mu_\pm - |u|^{q-1}u)$.

   • Unlike non-negative solutions where one deals with $u$, here the author handles the difference between the solution and its local supremum/infimum ($\mu_\pm$).
   • These inequalities bound the supremum of the integral of the "energy" of the solution in terms of its infimum plus a remainder term involving the cylinder dimensions.

3. Expansion of Positivity:
   The author proves a new theorem (Theorem 1.4) regarding the expansion of positivity for sign-changing solutions in the mixed cases. If a solution is "mostly" positive (or negative) on a set of significant measure at a specific time, it propagates to a larger set in a smaller cylinder, provided the solution is not too close to zero relative to the oscillation.

4. Reduction to Known Cases:

   • For Case A ($q < p-1, p < 2$), the problem is reduced to a singular parabolic $p$-Laplacian type behavior via the intrinsic scaling.
   • For Case B ($q > p-1, p > 2$), the problem is reduced to a singular doubly nonlinear equation.
   • The proof handles the "sign-changing" nature by splitting the analysis into cases where the solution is close to zero (requiring the new Harnack inequalities) and where it is bounded away from zero (allowing reduction to standard non-negative theory).

3. Key Contributions

• Extension of Regularity Theory: The paper extends the known ranges of parameters $(p, q)$ for which Hölder continuity holds to include the previously unstudied mixed degenerate/singular cases for sign-changing solutions.
• New Integral Inequalities: The derivation of Theorems 1.2 and 1.3 provides the first integral Harnack-type estimates for sign-changing solutions in these specific mixed regimes. These inequalities are crucial for controlling the oscillation of the solution without assuming a sign.
• Handling the Zero Level Set: The methodology explicitly addresses the difficulties arising when the solution crosses zero ($u=0$), a region where the equation's structure changes drastically. The use of the algebraic lemma (Lemma 2.1) allows the author to bound terms like $||a|^{q-1}a - |b|^{q-1}b|$ effectively.
• Unified Framework: The paper presents a unified approach using intrinsic cylinders $Q_{r, b \omega^{q-p+1} r^p}$ that adapts to both the $q < p-1$ and $q > p-1$ regimes.

4. Main Results

Theorem 1.1 (Main Result):
Let $u$ be a locally bounded, local weak solution to the equation. If the parameters satisfy either:

1. Mixed Degenerate/Singular: $0 < q < p-1$, $p < 2$, and $p + N(p-2) > 0$.
2. Mixed Singular/Degenerate: $q > p-1$, $p > 2$, and $qp + N(p-1-q) > 0$.

Then $u$ is locally Hölder continuous in $\Omega_T$.

Proposition 1.1 (Oscillation Decay):
The core of the proof establishes that for a solution $u$ in a cylinder defined by its oscillation $\omega$, the oscillation in a smaller, intrinsically scaled cylinder decays by a factor $\sigma_0 < 1$:
$$\text{osc}_{Q'} u \leq \sigma_0 \omega$$
This decay is achieved through a recursive argument involving the integral Harnack inequalities and the expansion of positivity.

Theorems 1.2 & 1.3 (Integral Harnack Inequalities):
These theorems provide bounds on the integral of the difference between the solution and its local extrema. They are the technical engine that drives the oscillation reduction, specifically handling the sign-changing nature of $u$ in the mixed regimes.

5. Significance

• Theoretical Completeness: This work fills a critical gap in the regularity theory of doubly nonlinear parabolic equations. Prior to this, the behavior of sign-changing solutions in mixed degenerate/singular regimes was an open problem.
• Methodological Innovation: The adaptation of DiBenedetto's intrinsic scaling to signed solutions, combined with the new integral Harnack inequalities, provides a robust toolkit for analyzing equations where the diffusion coefficient changes sign or vanishes.
• Applications: These equations model various physical phenomena, including non-Newtonian fluid flow, porous media flow with phase transitions, and filtration processes. Understanding the regularity of solutions that change sign (e.g., pressure differences or concentration gradients crossing zero) is essential for accurate physical modeling.
• Foundation for Future Work: The techniques developed here, particularly the handling of the zero level set in mixed regimes, can likely be applied to other nonlinear parabolic problems involving sign-changing data or coefficients.

In summary, Skrypnik successfully proves that despite the complex interplay of degeneracy and singularity, and the added difficulty of sign changes, the solutions to these doubly nonlinear equations remain locally Hölder continuous, provided the parameters fall within the specified mixed ranges.