Bounded solutions and interpolative gap bounds for degenerate parabolic double phase problems
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914447227879424 |
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| author | Kim, Bogi Oh, Jehan |
| author_facet | Kim, Bogi Oh, Jehan |
| contents | We establish gradient higher integrability results for weak solutions to degenerate parabolic equations of double phase type $$ u_t-\operatorname{div} \left(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du\right)=0 $$ in $Ω_T := Ω\times (0,T)$, where $a(\cdot)\in C^{α,\fracα{2}}(Ω_T)$. For bounded solutions, we prove that the result holds under the gap condition $$ q \leq p + α. $$ Moreover, for solutions with $$ u\in C(0,T;L^s(Ω)), \quad s \geq 2, $$ we obtain higher integrability under the gap condition $$ q \leq p + \frac{sα}{n+s}. $$ These results provide an interpolation between the gap bounds in the parabolic double phase setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_13454 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bounded solutions and interpolative gap bounds for degenerate parabolic double phase problems Kim, Bogi Oh, Jehan Analysis of PDEs Primary 35B65, Secondary 35K65, 35K55, 35D30 We establish gradient higher integrability results for weak solutions to degenerate parabolic equations of double phase type $$ u_t-\operatorname{div} \left(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du\right)=0 $$ in $Ω_T := Ω\times (0,T)$, where $a(\cdot)\in C^{α,\fracα{2}}(Ω_T)$. For bounded solutions, we prove that the result holds under the gap condition $$ q \leq p + α. $$ Moreover, for solutions with $$ u\in C(0,T;L^s(Ω)), \quad s \geq 2, $$ we obtain higher integrability under the gap condition $$ q \leq p + \frac{sα}{n+s}. $$ These results provide an interpolation between the gap bounds in the parabolic double phase setting. |
| title | Bounded solutions and interpolative gap bounds for degenerate parabolic double phase problems |
| topic | Analysis of PDEs Primary 35B65, Secondary 35K65, 35K55, 35D30 |
| url | https://arxiv.org/abs/2511.13454 |