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| Format: | Preprint |
| Veröffentlicht: |
2023
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2310.01048 |
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| _version_ | 1866915899949187072 |
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| author | Nadin, Grégoire |
| author_facet | Nadin, Grégoire |
| contents | We derive in this paper Gaussian estimates for a general parabolic equation $u_{t}-\big(a(x)u_{x}\big)_x= r(x)u$ over $\mathbb{R}$. Here $a$ and $r$ are only assumed to be bounded, measurable and $\mathrm{essinf}_{\mathbb{R}} a>0$. We first consider a canonical equation $ν(x) \partial_{t}p - \partial_{x }\big( ν(x)a(x)\partial_{x}p\big)+W\partial_{x}p=0$, with $W\in \mathbb{R}$, $ν$ bounded and $\mathrm{essinf}_{\mathbb{R}} ν>0$, for which we derive Gaussian estimates for the fundamental solution: $$\forall t>0, x,y\in \mathbb{R}, \quad \displaystyle\frac{1}{Ct^{1/2}}e^{-C|T(x)-T(y)-Wt|^{2}/t} \leq P(t,x,y)\leq \frac{C}{t^{1/2}}e^{-|T(x)-T(y)-Wt|^{2}/Ct}.$$ Here, the function $T$ is a corrector, for which we are able to derive appropriate properties using one-dimensional arguments. We then show that any solution $u$ of the original equation could be divided by some generalized principal eigenfunction $ϕ_γ$ so that $p:=u/ϕ_γ$ satisfies a canonical equation. As a byproduct of our proof, we derive Nash type estimates, that is, Holder continuity in $x$, for the solutions of the canonical equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_01048 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gaussian estimates for general parabolic operators in dimension 1 Nadin, Grégoire Analysis of PDEs We derive in this paper Gaussian estimates for a general parabolic equation $u_{t}-\big(a(x)u_{x}\big)_x= r(x)u$ over $\mathbb{R}$. Here $a$ and $r$ are only assumed to be bounded, measurable and $\mathrm{essinf}_{\mathbb{R}} a>0$. We first consider a canonical equation $ν(x) \partial_{t}p - \partial_{x }\big( ν(x)a(x)\partial_{x}p\big)+W\partial_{x}p=0$, with $W\in \mathbb{R}$, $ν$ bounded and $\mathrm{essinf}_{\mathbb{R}} ν>0$, for which we derive Gaussian estimates for the fundamental solution: $$\forall t>0, x,y\in \mathbb{R}, \quad \displaystyle\frac{1}{Ct^{1/2}}e^{-C|T(x)-T(y)-Wt|^{2}/t} \leq P(t,x,y)\leq \frac{C}{t^{1/2}}e^{-|T(x)-T(y)-Wt|^{2}/Ct}.$$ Here, the function $T$ is a corrector, for which we are able to derive appropriate properties using one-dimensional arguments. We then show that any solution $u$ of the original equation could be divided by some generalized principal eigenfunction $ϕ_γ$ so that $p:=u/ϕ_γ$ satisfies a canonical equation. As a byproduct of our proof, we derive Nash type estimates, that is, Holder continuity in $x$, for the solutions of the canonical equation. |
| title | Gaussian estimates for general parabolic operators in dimension 1 |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2310.01048 |