Barriers, Barenblatt solutions and regularity of soda can domains for the heat equation and nonlinear $p$-parabolic equations

Abstract: In this paper we study when the origin $(0,0)$ is a regular (or irregular) boundary point for the so-called soda can domains of the type [ Θ{l,θ}:= {(x,t) \in \mathbf{R}^{n+1}: 0<-t < θ|x|^l <θ}, \quad \text{with $l,θ>0$,} ] for the $p$-parabolic equation $\partial_t u- Δ_p u=0$, where $1<p<\infty$. For $p<2n/(n+1)$ and for the heat equation (i.e.\ $p=2$) we completely determine when the origin is regular for soda can domains. The domains $Θ{l,θ}$ have nonconvex time sections with power dependence on time. For domains with rotationally symmetric convex time sections with power dependence on time, the regularity of the origin as the last point was characterized by Petrovskii (in 1935) for the heat equation, and almost completely in the nonlinear case ($p \ne 2$) in our earlier paper (joint with Gianazza, Math. Ann. 368 (2017), 885--904).

• The paper introduces explicit barrier constructions and Barenblatt-type solutions to characterize the regularity at the origin in soda can domains.
• It demonstrates that regularity depends critically on parameters p, l, and the spatial dimension, revealing different regimes for 1 < p < 2 and p > 2.
• The study employs capacity estimates and Wiener-type tests to refine classical criteria, offering both complete and partial regularity results.

Boundary Regularity in Soda Can Domains for Heat and Nonlinear $p$-Parabolic Equations

Introduction

This paper (2604.27516) investigates the boundary regularity of solutions to the heat equation and to nonlinear $p$-parabolic equations on a novel class of time-dependent domains, termed "soda can domains." The study provides refined and, in certain regimes, full criteria for when the origin—interpreted as the last boundary point—is regular or irregular for the Dirichlet problem in such domains. The analysis leverages barriers, explicit Barenblatt-type solutions, and classical Wiener criterion techniques, revealing significant distinctions depending on the range of the parameter $p$. The results both extend and contrast earlier criteria such as those of Petrovskii and others for convex and nonconvex domains.

Soda Can Domains and the $p$-Parabolic Framework

Soda can domains are defined as

$$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$

parametrized by $l, \theta > 0$, and can display highly nonconvex time sections. The primary PDE of interest is the $p$-parabolic equation,

$\partial_t u - \Delta_p u = 0,$

where $$\Delta_p u = \nabla \cdot (|\nabla u|^{p-2} \nabla u)$$ and $1 < p < \infty$. The classical heat equation corresponds to $p$0.

While boundary regularity for domains with convex time sections is well-understood via criteria such as that of Petrovskii (for $p$1) and extending results for the $p$2-parabolic case, the scenario for nonconvex sections, particularly soda can domains, has not been comprehensively analyzed prior to this investigation.

Main Results on Regularity and Irregularity

The authors deliver a comprehensive characterization of the regularity of the origin for soda can domains in multiple settings:

Regularity Results

• For $p$3: If $p$4, the origin is regular for every $p$5 in $p$6.
• For $p$7: If $p$8, the origin is regular for every $p$9; special treatment is provided for the critical and supercritical regimes ($p$0 and $p$1).
• For $p$2: The origin is always regular, independent of $p$3 and $p$4, reflecting that the Barenblatt solutions provide explicit barriers in these highly nonlinear or low-dimensional regimes.
• For $p$5 (Heat Equation):
  • In $p$6, the origin is regular for all $p$7,
  • In $p$8, the origin is regular if and only if $p$9.

These regularity results are achieved by constructing explicit barrier families—either of power type or using Barenblatt's fundamental solutions. Notably, the independence of regularity from the scaling parameter $p$0 for $p$1 is rigorously shown via variable scaling arguments.

Irregularity Results

• For $p$2: If $p$3 (when $p$4) or $p$5 (for $p$6), the origin is irregular for all $p$7.
• For the heat equation in $p$8: If $p$9, the origin is irregular.
• For the punctured cylinder in $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$0: The origin is irregular, in sharp contrast to the result for soda can domains.

The critical exponent $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$1 appears as a threshold dictating the transition between the various regularity regimes. For $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$2, sharp regularity thresholds are established via the Wiener criterion, making rigorous use of parabolic capacity estimates.

Partial and Small Data Regularity for Borderline Regimes

For borderline cases, such as $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$3 or $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$4 when $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$5, regularity can only be deduced for sufficiently small or regular boundary data, but irregularity for large boundary data remains unresolved, highlighting an open dichotomy.

Methods: Barriers, Barenblatt Solutions, and Capacity Estimates

Barriers and Barenblatt Solutions

The methodology hinges on constructing barrier functions. For $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$6 and $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$7, radial power barriers are explicit supersolutions. For the entire supercritical and critical regimes, Barenblatt-type self-similar solutions serve as refined barriers, with precise identification of constants ensuring the barrier property is met. At the heart of the analysis is the observation that for $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$8, the PDE's homogeneity allows scaling to connect domains of various $$\Theta_{l, \theta} = \left\{(x, t) \in \mathbb{R}^{n+1} : 0 < -t < \theta |x|^l \right\},$$9, simplifying many proofs.

Capacity and Wiener-Type Tests

For $l, \theta > 0$0, capacity methods and the Wiener criterion are central. The paper includes detailed derivations and estimates of parabolic (thermal) capacities for cylinders, which feed into the Wiener test to distinguish regular and irregular boundary points. In $l, \theta > 0$1, the divergence of the harmonic series implies regularity for all domains of the given form, whereas for higher dimensions, the capacity exponent imposes stricter requirements.

Implications, Contradictory Features, and Open Problems

Theoretical Insights

This work reveals pronounced differences between the linear ($l, \theta > 0$2) and nonlinear ($l, \theta > 0$3) regimes:

• For $l, \theta > 0$4, the regularity at the origin is independent of the scaling parameter $l, \theta > 0$5; this strong claim is not mirrored for $l, \theta > 0$6. Moreover, for $l, \theta > 0$7, the critical exponents for regularity shift and the geometry of the domain plays a distinct role due to the nonlinearity.
• For $l, \theta > 0$8, new irregularity phenomena occur, and for $l, \theta > 0$9, regularity is "automatic" for all soda can domains. This is a contradicting behavior compared to the heat equation.
• Barrier families: Existence of a single barrier does not always imply regularity for nonlinear $p$0-parabolic equations with $p$1, in sharp contrast to the linear case and to nonlinear elliptic equations.

Practical and Applied Aspects

Soda can domains model media with time-dependent "holes" or inclusions—for instance, diffusion from shrinking sources or heat transfer problems with moving boundaries. The results clarify when solutions to related evolutionary PDEs are continuous up to the last point and inform numerical and analytical boundary treatment in these time-dependent nonconvex geometries.

Open Problems

Several open questions remain, particularly regarding:

• Regularity at the origin for $p$2 and $p$3, especially for large boundary data;
• Existence of barrier families for $p$4, $p$5;
• Sharp characterization for generalized, non-power type soda can domains, especially in $p$6.

Conclusion

This paper advances the theory of regularity for parabolic boundary value problems in nonconvex, time-varying domains, offering both complete and partial criteria depending on the nonlinear parameter and dimension. The classification is achieved via explicit barrier and Barenblatt-type constructions for $p$7-parabolic equations and via capacity-based Wiener tests for the heat equation. The results expose key distinctions between linear and nonlinear diffusion and open several avenues for further development in nonlinear potential theory and the analysis of noncylindrical evolutionary domains.