Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2602.12046 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910020529029120 |
|---|---|
| author | Giannetti, Flavia di Napoli, Antonia Passarelli Scheven, Christoph |
| author_facet | Giannetti, Flavia di Napoli, Antonia Passarelli Scheven, Christoph |
| contents | We investigate the local boundedness of solutions $u:Ω_T\to\mathbb{R}$ to parabolic equations of the form \begin{equation*}
\partial_tu-\mathrm{div}\,\mathcal{A}(x,t,Du)=0 \qquad\mbox{in }Ω_T=Ω\times(0,T) \end{equation*} that satisfy $p,q$-growth conditions and have degenerate coefficients. More precisely, we assume structure conditions of the type \begin{align*} |\mathcal{A}(x,t,ξ)|&\le b(x,t)(μ^2+|ξ|^2)^{\frac{q-1}{2}},\\ \langle \mathcal{A}(x,t,ξ),ξ\rangle&\ge a(x,t)(μ^2+|ξ|^2)^{\frac {p-2}{2}}|ξ|^2, \end{align*} for $2\le p\le q$ and $μ\in[0,1]$, where the functions $a^{-1}, b:Ω_T\to\mathbb{R}$ are possibly unbounded and only satisfy some integrability condition. Under a certain assumption on the gap between $p$ and $q$, we prove two main results. First, we show that subsolutions that are contained in the natural energy space are locally bounded from above. Second, for parabolic equations with a variational structure, we use these bounds to show the existence of locally bounded variational solutions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_12046 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local boundedness for solutions to parabolic $p,q$-problems with degenerate coefficients Giannetti, Flavia di Napoli, Antonia Passarelli Scheven, Christoph Analysis of PDEs We investigate the local boundedness of solutions $u:Ω_T\to\mathbb{R}$ to parabolic equations of the form \begin{equation*} \partial_tu-\mathrm{div}\,\mathcal{A}(x,t,Du)=0 \qquad\mbox{in }Ω_T=Ω\times(0,T) \end{equation*} that satisfy $p,q$-growth conditions and have degenerate coefficients. More precisely, we assume structure conditions of the type \begin{align*} |\mathcal{A}(x,t,ξ)|&\le b(x,t)(μ^2+|ξ|^2)^{\frac{q-1}{2}},\\ \langle \mathcal{A}(x,t,ξ),ξ\rangle&\ge a(x,t)(μ^2+|ξ|^2)^{\frac {p-2}{2}}|ξ|^2, \end{align*} for $2\le p\le q$ and $μ\in[0,1]$, where the functions $a^{-1}, b:Ω_T\to\mathbb{R}$ are possibly unbounded and only satisfy some integrability condition. Under a certain assumption on the gap between $p$ and $q$, we prove two main results. First, we show that subsolutions that are contained in the natural energy space are locally bounded from above. Second, for parabolic equations with a variational structure, we use these bounds to show the existence of locally bounded variational solutions. |
| title | Local boundedness for solutions to parabolic $p,q$-problems with degenerate coefficients |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.12046 |