On $α$-entropy solutions of a nonlocal thin film equation: existence and finite speed of propagation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866915758078951424 |
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| author | Segatti, Antonio Taranets, Roman |
| author_facet | Segatti, Antonio Taranets, Roman |
| contents | We consider an initial-boundary value problem for a class of nonlocal thin film equations governed by the spectral fractional Laplacian with homogeneous Neumann boundary conditions. We were the first to establish an $α$-entropy estimate for nonlocal thin film equations, which yields essential a priori bounds for the regularity and long-time behavior of weak solutions. By developing a localized version of this estimate, we prove finite speed of propagation, showing that the support of nonnegative solutions remains compact for positive times. Furthermore, we find a sufficient condition for a waiting time phenomenon, whereby the solution remains identically zero in a region for a nontrivial time interval. These results highlight new features in the interaction between nonlocal effects and classical thin film dynamics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_19808 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $α$-entropy solutions of a nonlocal thin film equation: existence and finite speed of propagation Segatti, Antonio Taranets, Roman Analysis of PDEs We consider an initial-boundary value problem for a class of nonlocal thin film equations governed by the spectral fractional Laplacian with homogeneous Neumann boundary conditions. We were the first to establish an $α$-entropy estimate for nonlocal thin film equations, which yields essential a priori bounds for the regularity and long-time behavior of weak solutions. By developing a localized version of this estimate, we prove finite speed of propagation, showing that the support of nonnegative solutions remains compact for positive times. Furthermore, we find a sufficient condition for a waiting time phenomenon, whereby the solution remains identically zero in a region for a nontrivial time interval. These results highlight new features in the interaction between nonlocal effects and classical thin film dynamics. |
| title | On $α$-entropy solutions of a nonlocal thin film equation: existence and finite speed of propagation |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2601.19808 |