On well-posedness for non-autonomous parabolic Cauchy problems with rough initial data
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913836960841728 |
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| author | Hou, Hedong |
| author_facet | Hou, Hedong |
| contents | We establish a complete picture for existence, uniqueness, and representation of weak solutions to non-autonomous parabolic Cauchy problems of divergence type. The coefficients are only assumed to be uniformly elliptic, bounded, measurable, and complex-valued, without any additional regularity or symmetry conditions. The initial data are tempered distributions taken in homogeneous Hardy--Sobolev spaces $\dot{H}^{s,p}$, and source terms belong to certain scales of weighted tent spaces. Weak solutions are constructed with their gradients in weighted tent spaces $T^{p}_{s/2}$. Analogous results are also exhibited for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09387 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On well-posedness for non-autonomous parabolic Cauchy problems with rough initial data Hou, Hedong Analysis of PDEs Classical Analysis and ODEs Primary 35K15, Secondary 42B37, 35B45, 42B30, 46E35 We establish a complete picture for existence, uniqueness, and representation of weak solutions to non-autonomous parabolic Cauchy problems of divergence type. The coefficients are only assumed to be uniformly elliptic, bounded, measurable, and complex-valued, without any additional regularity or symmetry conditions. The initial data are tempered distributions taken in homogeneous Hardy--Sobolev spaces $\dot{H}^{s,p}$, and source terms belong to certain scales of weighted tent spaces. Weak solutions are constructed with their gradients in weighted tent spaces $T^{p}_{s/2}$. Analogous results are also exhibited for initial data in homogeneous Besov spaces $\dot{B}^{s}_{p,p}$. |
| title | On well-posedness for non-autonomous parabolic Cauchy problems with rough initial data |
| topic | Analysis of PDEs Classical Analysis and ODEs Primary 35K15, Secondary 42B37, 35B45, 42B30, 46E35 |
| url | https://arxiv.org/abs/2505.09387 |