Quantitative and exact concavity principles for parabolic and elliptic equations
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arXiv
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| Formato: | Preprint |
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2025
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| _version_ | 1866917048634834944 |
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| author | Gallo, Marco Moraschi, Riccardo Squassina, Marco |
| author_facet | Gallo, Marco Moraschi, Riccardo Squassina, Marco |
| contents | Goal of this paper is to study classes of Cauchy-Dirichlet problems which include parabolic equations of the type $$u_t -Δu= a(x,t)f(u)\quad\hbox{in $Ω\times(0,T)$}$$ with $Ω\subset\mathbb{R}^N$ bounded, convex domain and $T\in(0,+\infty]$. Under suitable assumptions on $a$ and $f$, we show logarithmic or power concavity (in space, or in space-time) of the solution $u$; under some relaxed assumptions on $a$, we show moreover that $u$ enjoys concavity properties up to a controlled error. The results include relevant examples like the torsion $f(u)=1$, the Lane-Emden equation $f(u)=u^q$, $q\in(0,1)$, the eigenfunction $f(u)=u$, the logarithmic equation $f(u)=u\log(u^2)$, and the saturable nonlinearity $f(u)=\frac{u^2}{1+u}$. The logistic equation $f(x,u)=a(x)u-u^2$ can be treated as well.
Some exact results give a different approach, as well as generalizations, to [Ishige-Salani2013, Ishige-Salani2016]. Moreover, some quantitative results are valid also in the elliptic framework $-Δu=a(x)f(u)$ and refine [Bucur-Squassina2019, Gallo-Squassina2024]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_09494 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative and exact concavity principles for parabolic and elliptic equations Gallo, Marco Moraschi, Riccardo Squassina, Marco Analysis of PDEs 26B25, 35B09, 26B25, 35B99, 35E10, 35J60, 35K20, 35K91 Goal of this paper is to study classes of Cauchy-Dirichlet problems which include parabolic equations of the type $$u_t -Δu= a(x,t)f(u)\quad\hbox{in $Ω\times(0,T)$}$$ with $Ω\subset\mathbb{R}^N$ bounded, convex domain and $T\in(0,+\infty]$. Under suitable assumptions on $a$ and $f$, we show logarithmic or power concavity (in space, or in space-time) of the solution $u$; under some relaxed assumptions on $a$, we show moreover that $u$ enjoys concavity properties up to a controlled error. The results include relevant examples like the torsion $f(u)=1$, the Lane-Emden equation $f(u)=u^q$, $q\in(0,1)$, the eigenfunction $f(u)=u$, the logarithmic equation $f(u)=u\log(u^2)$, and the saturable nonlinearity $f(u)=\frac{u^2}{1+u}$. The logistic equation $f(x,u)=a(x)u-u^2$ can be treated as well. Some exact results give a different approach, as well as generalizations, to [Ishige-Salani2013, Ishige-Salani2016]. Moreover, some quantitative results are valid also in the elliptic framework $-Δu=a(x)f(u)$ and refine [Bucur-Squassina2019, Gallo-Squassina2024]. |
| title | Quantitative and exact concavity principles for parabolic and elliptic equations |
| topic | Analysis of PDEs 26B25, 35B09, 26B25, 35B99, 35E10, 35J60, 35K20, 35K91 |
| url | https://arxiv.org/abs/2504.09494 |