Heat and wave type equations with non-local operators, I. Compact Lie groups
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866917577512452096 |
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| author | de Moraes, Wagner A. A. Restrepo, Joel E. Ruzhansky, Michael |
| author_facet | de Moraes, Wagner A. A. Restrepo, Joel E. Ruzhansky, Michael |
| contents | We prove existence, uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a non-local (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty$. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat type equation as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_05608 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Heat and wave type equations with non-local operators, I. Compact Lie groups de Moraes, Wagner A. A. Restrepo, Joel E. Ruzhansky, Michael Analysis of PDEs 22C05, 45K05, 35B40 We prove existence, uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a non-local (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty$. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat type equation as well. |
| title | Heat and wave type equations with non-local operators, I. Compact Lie groups |
| topic | Analysis of PDEs 22C05, 45K05, 35B40 |
| url | https://arxiv.org/abs/2210.05608 |