Global existence and asymptotic behavior of the Westervelt--hyperbolic Pennes system
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916194935635968 |
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| author | Benabbas, Imen Said-Houari, Belkacem |
| author_facet | Benabbas, Imen Said-Houari, Belkacem |
| contents | In this work, we investigate the global existence and asymptotic behavior of a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt--Pennes--Cattaneo model). First, we prove that the solution exists globally in time, provided that the lower-order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This is done using a continuity argument together with some interpolation inequalities. Second, we prove an exponential decay of the solution under the same smallness assumptions on the initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03920 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Global existence and asymptotic behavior of the Westervelt--hyperbolic Pennes system Benabbas, Imen Said-Houari, Belkacem Analysis of PDEs In this work, we investigate the global existence and asymptotic behavior of a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt--Pennes--Cattaneo model). First, we prove that the solution exists globally in time, provided that the lower-order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This is done using a continuity argument together with some interpolation inequalities. Second, we prove an exponential decay of the solution under the same smallness assumptions on the initial data. |
| title | Global existence and asymptotic behavior of the Westervelt--hyperbolic Pennes system |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.03920 |