Generalized exponential pullback attractor for a nonautonomous wave equation
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910295354507264 |
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| author | Bortolan, Matheus C. Caraballo, Tomas Neto, Carlos Pecorari |
| author_facet | Bortolan, Matheus C. Caraballo, Tomas Neto, Carlos Pecorari |
| contents | In this work we introduce the concept of generalized exponential $\mathfrak{D}$-pullback attractor for evolution processes, where $\mathfrak{D}$ is a universe of families in $X$, which is a compact and positively invariant family that pullback attracts all elements of $\mathfrak{D}$ with an exponential rate. Such concept was introduced in arXiv:2311.15630 for the general case of decaying functions (which include the exponential decay), but for fixed bounded sets rather than to universe of families. We prove a result that ensures the existence of a generalized exponential $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor for an evolution process, where $\mathfrak{D}_{\mathcal{C}^\ast}$ is a specific universe. This required an adaptation of the results of arXiv:2311.15630, which only covered the case of a polynomial rate of attraction, for fixed bounded sets. Later, we prove that a nonautonomous wave equation has a generalized exponential $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor. This, in turn, also implies the existence of the $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor for such problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06631 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalized exponential pullback attractor for a nonautonomous wave equation Bortolan, Matheus C. Caraballo, Tomas Neto, Carlos Pecorari Dynamical Systems Analysis of PDEs 35B41, 35L20, 37L25 In this work we introduce the concept of generalized exponential $\mathfrak{D}$-pullback attractor for evolution processes, where $\mathfrak{D}$ is a universe of families in $X$, which is a compact and positively invariant family that pullback attracts all elements of $\mathfrak{D}$ with an exponential rate. Such concept was introduced in arXiv:2311.15630 for the general case of decaying functions (which include the exponential decay), but for fixed bounded sets rather than to universe of families. We prove a result that ensures the existence of a generalized exponential $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor for an evolution process, where $\mathfrak{D}_{\mathcal{C}^\ast}$ is a specific universe. This required an adaptation of the results of arXiv:2311.15630, which only covered the case of a polynomial rate of attraction, for fixed bounded sets. Later, we prove that a nonautonomous wave equation has a generalized exponential $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor. This, in turn, also implies the existence of the $\mathfrak{D}_{\mathcal{C}^\ast}$-pullback attractor for such problem. |
| title | Generalized exponential pullback attractor for a nonautonomous wave equation |
| topic | Dynamical Systems Analysis of PDEs 35B41, 35L20, 37L25 |
| url | https://arxiv.org/abs/2401.06631 |