Barriers, Barenblatt solutions and regularity of soda can domains for the heat equation and nonlinear $p$-parabolic equations
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arXiv
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2026
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| author | Björn, Anders Björn, Jana |
| author_facet | Björn, Anders Björn, Jana |
| contents | 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). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_27516 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Barriers, Barenblatt solutions and regularity of soda can domains for the heat equation and nonlinear $p$-parabolic equations Björn, Anders Björn, Jana Analysis of PDEs Primary: 35K20, Secondary: 35K05, 35K65, 35K67, 35K92 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). |
| title | Barriers, Barenblatt solutions and regularity of soda can domains for the heat equation and nonlinear $p$-parabolic equations |
| topic | Analysis of PDEs Primary: 35K20, Secondary: 35K05, 35K65, 35K67, 35K92 |
| url | https://arxiv.org/abs/2604.27516 |