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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2310.08871 |
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| _version_ | 1866929514049699840 |
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| author | Kang, Jaehoon Park, Daehan |
| author_facet | Kang, Jaehoon Park, Daehan |
| contents | In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators $$ \partial_{t}u(t,x) = \mathcal{L}^{a}u(t,x) + f(t,x), \quad t>0 $$ in $L_{q}(L_{p})$ spaces. Our spatial operator $\mathcal{L}^{a}$ is an integro-differential operator of the form $$ \int_{\mathbb{R}^{d}} \left( u(x+y)-u(x) -\nabla u(x) \cdot y \mathrm{1}_{|y|\leq 1} \right) a(t,y) j_{d}(|y|)dy. $$ Here, $a(t,y)$ is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on $j_{d}(r)$ which yield $L_{q}(L_{p})$-regularity of solutions. Our assumptions on $j_d$ are general so that $j_d(r)$ may be comparable to $r^{-d}\ell(r^{-1})$ for a function $\ell$ which is slowly varying at infinity. For example, we can take $\ell(r)=\log{(1+r^α)}$ or $\ell(r) = \min{\{r^α,1\}}$ ($α\in(0,2)$). Indeed, our result covers the operators whose Fourier multiplier $ψ(ξ)$ does not have any scaling condition for $|ξ|\geq 1$. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on $ψ$ are considered. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_08871 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | An $L_{q}(L_{p})$-regularity theory for parabolic equations with integro-differential operators having low intensity kernels Kang, Jaehoon Park, Daehan Analysis of PDEs Probability 35B65, 47G20, 60G51, 60J35 In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators $$ \partial_{t}u(t,x) = \mathcal{L}^{a}u(t,x) + f(t,x), \quad t>0 $$ in $L_{q}(L_{p})$ spaces. Our spatial operator $\mathcal{L}^{a}$ is an integro-differential operator of the form $$ \int_{\mathbb{R}^{d}} \left( u(x+y)-u(x) -\nabla u(x) \cdot y \mathrm{1}_{|y|\leq 1} \right) a(t,y) j_{d}(|y|)dy. $$ Here, $a(t,y)$ is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on $j_{d}(r)$ which yield $L_{q}(L_{p})$-regularity of solutions. Our assumptions on $j_d$ are general so that $j_d(r)$ may be comparable to $r^{-d}\ell(r^{-1})$ for a function $\ell$ which is slowly varying at infinity. For example, we can take $\ell(r)=\log{(1+r^α)}$ or $\ell(r) = \min{\{r^α,1\}}$ ($α\in(0,2)$). Indeed, our result covers the operators whose Fourier multiplier $ψ(ξ)$ does not have any scaling condition for $|ξ|\geq 1$. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on $ψ$ are considered. |
| title | An $L_{q}(L_{p})$-regularity theory for parabolic equations with integro-differential operators having low intensity kernels |
| topic | Analysis of PDEs Probability 35B65, 47G20, 60G51, 60J35 |
| url | https://arxiv.org/abs/2310.08871 |