Well-posedness and regularity for seminlinear time-dependent second and fourth order in space equations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866918492363554816 |
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| author | Remesan, Gopikrishnan Chirappurathu |
| author_facet | Remesan, Gopikrishnan Chirappurathu |
| contents | This article discusses a unified convergence analysis of the semilinear time-dependent equation $\partial_t u + (-1)^\mathrm{m}Δ^{\mathrm{m}}u + u^3 - u = f$ with $\mathrm{m} \in \{1,2\}$ and homogeneous Dirichlet boundary conditions. The analysis relies on Faedo-Galerkin approximation and convergence via compactness estimates. The existence and uniqueness of the weak solution is proved when the initial data is smooth. A refined and novel analysis extends the existence result to problems with rough initial data also. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08714 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Well-posedness and regularity for seminlinear time-dependent second and fourth order in space equations Remesan, Gopikrishnan Chirappurathu Analysis of PDEs 35K58, 35K91, 35K35 (Primary), 46B50 (Secondary) This article discusses a unified convergence analysis of the semilinear time-dependent equation $\partial_t u + (-1)^\mathrm{m}Δ^{\mathrm{m}}u + u^3 - u = f$ with $\mathrm{m} \in \{1,2\}$ and homogeneous Dirichlet boundary conditions. The analysis relies on Faedo-Galerkin approximation and convergence via compactness estimates. The existence and uniqueness of the weak solution is proved when the initial data is smooth. A refined and novel analysis extends the existence result to problems with rough initial data also. |
| title | Well-posedness and regularity for seminlinear time-dependent second and fourth order in space equations |
| topic | Analysis of PDEs 35K58, 35K91, 35K35 (Primary), 46B50 (Secondary) |
| url | https://arxiv.org/abs/2605.08714 |