Local boundedness for solutions to degenerate parabolic double phase problems

Abstract: In this paper, we investigate the local boundedness of weak solutions to degenerate parabolic double phase equation of type $$ u_t-\textrm{div}(|Du|^{p-2}Du+a(x,t)|Du|^{q-2}Du)=0\quad \text{in } Ω_T := Ω\times (0,T), $$ where $0\leq a(\cdot)\in L^\infty(Ω_T)$. To this end, we derive the Caccioppoli inequality and a parabolic embedding theorem, which are then utilized in an iteration method.

• The paper proves that weak solutions to degenerate parabolic double phase equations are locally bounded under minimal L∞ assumptions on the modulating coefficient.
• The paper adapts the De Giorgi iteration framework by developing a tailored Caccioppoli inequality and a novel parabolic embedding lemma to control the supremum in nested cylinders.
• The paper identifies a sharp gap condition between the exponents p and q, which is critical for ensuring regularity in non-uniformly parabolic settings and advancing previous approaches.

Local Boundedness for Solutions to Degenerate Parabolic Double Phase Problems

Introduction and Problem Formulation

The paper "Local boundedness for solutions to degenerate parabolic double phase problems" (2604.14544) addresses the local boundedness of weak solutions to parabolic PDEs with double phase structure, which can be written as

$$u_t - \mathrm{div} (|Du|^{p-2} Du + a(x,t)|Du|^{q-2} Du) = 0,$$

in $\Omega_T = \Omega \times (0,T)$, for $2 \leq p < q < p+\frac{p}{n}$ and $a(\cdot) \in L^\infty(\Omega_T)$ non-negative. Double phase problems are prototypical non-uniformly elliptic (or parabolic) equations stemming from energy functionals which possess $(p,q)$-growth, discovering applications in nonlinear elasticity, electrorheology, and image analysis. The analysis is fundamentally complicated by the interaction between genuinely degenerate $(p,q)$-growth and the low regularity of $a(x,t)$, with classical tools failing outside uniformly elliptic scenarios or when $a$ lacks sufficient regularity.

The authors focus on weak solutions in the natural energy space $C(0,T;L^2(\Omega)) \cap L^q(0,T;W^{1,q}(\Omega))$, for which existence and higher integrability have been recently established under more restrictive conditions [Wontae2023a, 2023_Gradient_Higher_Integrability_for_Degenerate_Parabolic_Double-Phase_Systems]. The main innovation is proving local boundedness solely under the minimal integrability $a \in L^\infty$ and an explicit, sharp gap bound on $\Omega_T = \Omega \times (0,T)$0, bypassing the need for Hölder regularity of $\Omega_T = \Omega \times (0,T)$1 that had previously been required. This places the local boundedness result essentially at the critical threshold for regularity theory in this setting.

Main Results and Techniques

The central theorem asserts that under the conditions $\Omega_T = \Omega \times (0,T)$2 and $\Omega_T = \Omega \times (0,T)$3, every weak solution $\Omega_T = \Omega \times (0,T)$4 is locally bounded, with a quantified estimate for the supremum in parabolic cylinders. The proof strategy is notable for refining the De Giorgi iteration framework to the degenerate, non-uniformly parabolic setting with double phase structure:

• Caccioppoli Inequality: The authors establish an energy estimate (Caccioppoli-type) adapted to the $\Omega_T = \Omega \times (0,T)$5-structure, which is delicate due to the competition between the $\Omega_T = \Omega \times (0,T)$6 and $\Omega_T = \Omega \times (0,T)$7 terms. The proof uses Steklov averaging and exploits the Carathéodory structure of the vector field.
• Parabolic Embedding: A new embedding lemma is shown, providing for $\Omega_T = \Omega \times (0,T)$8 and $\Omega_T = \Omega \times (0,T)$9, an estimate of higher $2 \leq p < q < p+\frac{p}{n}$0 norms in terms of $2 \leq p < q < p+\frac{p}{n}$1 norms and $2 \leq p < q < p+\frac{p}{n}$2 norms, with the key technical point being the threshold $2 \leq p < q < p+\frac{p}{n}$3.
• Iteration Scheme: Direct iteration using the previous two ingredients yields a decay estimate on the measure of super-level sets of $2 \leq p < q < p+\frac{p}{n}$4 in nested parabolic cylinders. At the critical step, the explicit dependence of the iteration exponent $2 \leq p < q < p+\frac{p}{n}$5 on the gap $2 \leq p < q < p+\frac{p}{n}$6 and spatial dimension ensures the sequence converges, thereby controlling the supremum.
• Regularity Consequences: The local boundedness is shown to be sufficient to deduce local Hölder continuity under additional regularity of the coefficient, thus connecting the present result to the optimal regularity theory known for elliptic double phase functionals.

The principal estimate for local boundedness has the form

$2 \leq p < q < p+\frac{p}{n}$7

with exponents and $2 \leq p < q < p+\frac{p}{n}$8 depending on the structural parameters and $2 \leq p < q < p+\frac{p}{n}$9, and where $a(\cdot) \in L^\infty(\Omega_T)$0 is the precise threshold established by the embedding.

Context and Comparative Discussion

The result provides the sharp local boundedness threshold for parabolic double phase equations with only $a(\cdot) \in L^\infty(\Omega_T)$1 control on the modulating coefficient, extending the classical De Giorgi-Nash-Moser theory—in which $a(\cdot) \in L^\infty(\Omega_T)$2-Laplace type equations admit uniform local bounds—to problems where non-uniform degeneracy and singularity manifest. Previous approaches [Wontae2023a, 2023_Gradient_Higher_Integrability_for_Degenerate_Parabolic_Double-Phase_Systems, kim2025boundedsolutionsinterpolativegap] necessitated additional Hölder continuity of $a(\cdot) \in L^\infty(\Omega_T)$3 or more restrictive structural gap bounds. The analysis here identifies the precise role played by the gap parameter, showing unique behavior compared to the elliptic case, where the optimal range is $a(\cdot) \in L^\infty(\Omega_T)$4. The techniques also avoid any reliance on the Lavrentiev phenomenon and relate to the sharpness underlying the lack of regularity for higher $a(\cdot) \in L^\infty(\Omega_T)$5.

Further, the integration of the result with the parabolic regularity program clarifies the connection between boundedness, higher integrability, and continuity. It completes a crucial step: the proof that the higher order regularity results (e.g., local Hölder estimates) in [Wontae2025] rely primarily on this optimal boundedness estimate.

Implications and Future Directions

The conclusions have substantial implications for the regularity theory of nonlinear PDEs with non-uniform growth and lead to several avenues for advancement:

• Sharpness and Limitations: The gap condition $a(\cdot) \in L^\infty(\Omega_T)$6 is shown to be sharp for the double phase scenario. The result draws a clear dichotomy in solution behavior, with potential unboundedness or loss of regularity when $a(\cdot) \in L^\infty(\Omega_T)$7 exceeds the threshold.
• Extensions: The techniques can be adapted to analyze equations with more complex modulation (e.g., two modulating coefficients [Kim2026]), to singular double phase cases [kim2026interpolativerefinementgapbound], or to systems with variable exponent growth.
• Applications: Practical contexts such as image reconstruction and nonlinear mechanics, which motivate double phase models, may benefit from rigorous regularity and boundedness guarantees under minimal a priori regularity on the coefficients.
• Open Problems: New directions include extending boundedness to systems, lower regularity for $a(\cdot) \in L^\infty(\Omega_T)$8, and understanding subtle phenomena for the borderline case $a(\cdot) \in L^\infty(\Omega_T)$9. Moreover, the techniques may provide insight into the fine structure of the Lavrentiev phenomenon [kim2026absencelavrentievphenomenondegenerate].

Conclusion

The paper makes a significant technical contribution by establishing local boundedness for weak solutions of degenerate parabolic double phase equations under the minimal gap bound and without invoking strong regularity on the modulating coefficient. The result clarifies the threshold for local regularity in non-uniformly parabolic PDEs of $(p,q)$0-type and serves as a foundation for the development of global regularity theories in this context, informing both the mathematical theory and practical applications of double phase models (2604.14544).