Interpolative Refinement of Gap Bound Conditions for Singular Parabolic Double Phase Problems
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910104314445824 |
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| author | Kim, Bogi Oh, Jehan |
| author_facet | Kim, Bogi Oh, Jehan |
| contents | We consider inhomogeneous singular parabolic double phase equations of type $$ u_t-\operatorname{div}(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du)=-\operatorname{div} (|F|^{p-2}F + a(x,t)|F|^{q-2}F) $$ in $Ω_T := Ω\times (0,T)\subset \mathbb{R}^n\times \mathbb{R}$, where $\frac{2n}{n+2}<p\leq 2$, $p<q$ and $0\leq a(\cdot)\in C^{α,\fracα{2}}(Ω_T)$. We establish gradient higher integrability results for weak solutions to the above problems under one of the following two assumptions: $$ u\in L^\infty (Ω_T) \quad\text{and}\quad q\leq p +\frac{α(p(n+2)-2n)}{4}, $$ or $$ u\in C(0,T;L^s(Ω)),\quad s\geq 2 \quad\text{and}\quad q\leq p+\frac{αμ_s}{n+s}, $$ where $μ_s := \frac{(p(n+2)-2n)s}{4}$. These results yield an interpolation refinement of gap bounds in the singular parabolic double phase setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01571 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Interpolative Refinement of Gap Bound Conditions for Singular Parabolic Double Phase Problems Kim, Bogi Oh, Jehan Analysis of PDEs Primary 35K67, Secondary 35D30, 35K55 We consider inhomogeneous singular parabolic double phase equations of type $$ u_t-\operatorname{div}(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du)=-\operatorname{div} (|F|^{p-2}F + a(x,t)|F|^{q-2}F) $$ in $Ω_T := Ω\times (0,T)\subset \mathbb{R}^n\times \mathbb{R}$, where $\frac{2n}{n+2}<p\leq 2$, $p<q$ and $0\leq a(\cdot)\in C^{α,\fracα{2}}(Ω_T)$. We establish gradient higher integrability results for weak solutions to the above problems under one of the following two assumptions: $$ u\in L^\infty (Ω_T) \quad\text{and}\quad q\leq p +\frac{α(p(n+2)-2n)}{4}, $$ or $$ u\in C(0,T;L^s(Ω)),\quad s\geq 2 \quad\text{and}\quad q\leq p+\frac{αμ_s}{n+s}, $$ where $μ_s := \frac{(p(n+2)-2n)s}{4}$. These results yield an interpolation refinement of gap bounds in the singular parabolic double phase setting. |
| title | Interpolative Refinement of Gap Bound Conditions for Singular Parabolic Double Phase Problems |
| topic | Analysis of PDEs Primary 35K67, Secondary 35D30, 35K55 |
| url | https://arxiv.org/abs/2601.01571 |