Anharmonic semigroups and applications to global well-posedness of nonlinear heat equations
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arXiv
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| Auteurs principaux: | , , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866914652406939648 |
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| author | Cardona, Duván Chatzakou, Marianna Delgado, Julio Kumar, Vishvesh Ruzhansky, Michael |
| author_facet | Cardona, Duván Chatzakou, Marianna Delgado, Julio Kumar, Vishvesh Ruzhansky, Michael |
| contents | In this work we consider the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ for $γ>0$ associated to an anharmonic oscillator of the form $ \mathcal{A}_{k,\,\ell}=(-Δ)^{\ell}+|x|^{2k}$ where $k,\ell$ are integers $\geq 1$. By introducing a suitable Hörmander metric on the phase-space we analyse the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ within the framework of Hörmander $S(M,g)$ classes and obtain mapping properties in the scale of modulation spaces $M^{p,q},\, 0<p,q\leq \infty,$ with respect to an anharmonic modulation weight. As an application, we apply the obtained bounds to establish the well-posedness for the nonlinear heat equation associated with $\mathcal{A}_{k,\,\ell}^γ$. It is worth noting that the results presented in this paper are novel, even in the case where $γ=1.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_13750 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Anharmonic semigroups and applications to global well-posedness of nonlinear heat equations Cardona, Duván Chatzakou, Marianna Delgado, Julio Kumar, Vishvesh Ruzhansky, Michael Analysis of PDEs In this work we consider the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ for $γ>0$ associated to an anharmonic oscillator of the form $ \mathcal{A}_{k,\,\ell}=(-Δ)^{\ell}+|x|^{2k}$ where $k,\ell$ are integers $\geq 1$. By introducing a suitable Hörmander metric on the phase-space we analyse the semigroup $e^{-t\mathcal{A}_{k,\,\ell}^γ}$ within the framework of Hörmander $S(M,g)$ classes and obtain mapping properties in the scale of modulation spaces $M^{p,q},\, 0<p,q\leq \infty,$ with respect to an anharmonic modulation weight. As an application, we apply the obtained bounds to establish the well-posedness for the nonlinear heat equation associated with $\mathcal{A}_{k,\,\ell}^γ$. It is worth noting that the results presented in this paper are novel, even in the case where $γ=1.$ |
| title | Anharmonic semigroups and applications to global well-posedness of nonlinear heat equations |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.13750 |