Regularity estimates to quasi-linear parabolic equations in non-divergence form with non-homogeneous signature
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910860049383424 |
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| author | Bessa, Junior da Silva da Silva, João Vitor Sá, Ginaldo de Santana |
| author_facet | Bessa, Junior da Silva da Silva, João Vitor Sá, Ginaldo de Santana |
| contents | In this manuscript, we establish the existence and sharp geometric regularity estimates for bounded solutions of a class of quasilinear parabolic equations in non-divergence form with non-homogeneous degeneracy. The model equation in this class is given by \[ \partial_{t} u = \left(|\nabla u|^{\mathfrak{p}} + \mathfrak{a}(x, t)|\nabla u|^{\mathfrak{q}}\right)Δ_{p}^{\mathrm{N}} u + f(x, t) \quad \text{in} \quad Q_1 = B_1 \times (-1, 0], \] where $p \in (1, \infty)$, $\mathfrak{p}, \mathfrak{q} \in [0, \infty)$, and $\mathfrak{a}, f: Q_1 \to \mathbb{R}$ are suitably defined functions. Our approach is based on geometric tangential methods, incorporating a refined oscillation mechanism, compactness arguments, ``alternative methods,'' and scaling techniques. Furthermore, we derive pointwise estimates in settings exhibiting singular-degenerate or doubly singular signatures. To some extent, our regularity estimates refine and extend previous results from \cite{FZ23} through distinct methodological advancements. Finally, we explore connections between our findings and fundamental nonlinear models in the theory of quasilinear PDEs, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03886 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity estimates to quasi-linear parabolic equations in non-divergence form with non-homogeneous signature Bessa, Junior da Silva da Silva, João Vitor Sá, Ginaldo de Santana Analysis of PDEs In this manuscript, we establish the existence and sharp geometric regularity estimates for bounded solutions of a class of quasilinear parabolic equations in non-divergence form with non-homogeneous degeneracy. The model equation in this class is given by \[ \partial_{t} u = \left(|\nabla u|^{\mathfrak{p}} + \mathfrak{a}(x, t)|\nabla u|^{\mathfrak{q}}\right)Δ_{p}^{\mathrm{N}} u + f(x, t) \quad \text{in} \quad Q_1 = B_1 \times (-1, 0], \] where $p \in (1, \infty)$, $\mathfrak{p}, \mathfrak{q} \in [0, \infty)$, and $\mathfrak{a}, f: Q_1 \to \mathbb{R}$ are suitably defined functions. Our approach is based on geometric tangential methods, incorporating a refined oscillation mechanism, compactness arguments, ``alternative methods,'' and scaling techniques. Furthermore, we derive pointwise estimates in settings exhibiting singular-degenerate or doubly singular signatures. To some extent, our regularity estimates refine and extend previous results from \cite{FZ23} through distinct methodological advancements. Finally, we explore connections between our findings and fundamental nonlinear models in the theory of quasilinear PDEs, which may be of independent interest. |
| title | Regularity estimates to quasi-linear parabolic equations in non-divergence form with non-homogeneous signature |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2503.03886 |