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| Autor principal: | |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2406.15684 |
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| _version_ | 1866913401650806784 |
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| author | Beceanu, Marius |
| author_facet | Beceanu, Marius |
| contents | This paper establishes the local exact controllability of the quasilinear porous media equation with Dirichlet boundary condition.\\ Consider the equation $$\begin{aligned} &y_t - Δa(y) = mu+f \text{ on } Q\\ &y(0)=y_0,\ y \mid_Σ = 0 \end{aligned}$$ on the $n+1$-dimensional cylinder $Q = Ω\times (0, T)$ with lateral boundary $Σ= \partial Ω\times (0, T)$. The exact controllability in finite time is proved when $\|y_0 - y_s\|_{W^{1, n}_0(Ω) \cap C(\overline Ω)}$ is sufficiently small, $n > 1$, for every stationary solution $y_s$ such that $a(y_s) \in W^{2, q}(Ω)$, where $q>n$. It is assumed that $Ω$ is a bounded open set with $C^2$ boundary and that $a \in C^2(\mathbb R)$, $a'>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_15684 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local Exact Controllability to Stationary Solutions of a Semilinear Parabolic Equation Beceanu, Marius Analysis of PDEs 93B05, 93C10, 93C35 This paper establishes the local exact controllability of the quasilinear porous media equation with Dirichlet boundary condition.\\ Consider the equation $$\begin{aligned} &y_t - Δa(y) = mu+f \text{ on } Q\\ &y(0)=y_0,\ y \mid_Σ = 0 \end{aligned}$$ on the $n+1$-dimensional cylinder $Q = Ω\times (0, T)$ with lateral boundary $Σ= \partial Ω\times (0, T)$. The exact controllability in finite time is proved when $\|y_0 - y_s\|_{W^{1, n}_0(Ω) \cap C(\overline Ω)}$ is sufficiently small, $n > 1$, for every stationary solution $y_s$ such that $a(y_s) \in W^{2, q}(Ω)$, where $q>n$. It is assumed that $Ω$ is a bounded open set with $C^2$ boundary and that $a \in C^2(\mathbb R)$, $a'>0$. |
| title | Local Exact Controllability to Stationary Solutions of a Semilinear Parabolic Equation |
| topic | Analysis of PDEs 93B05, 93C10, 93C35 |
| url | https://arxiv.org/abs/2406.15684 |