A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929369340968960 |
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| author | Iagar, Razvan Gabriel Munteanu, Diana Rodica |
| author_facet | Iagar, Razvan Gabriel Munteanu, Diana Rodica |
| contents | This is the first of a two-parts work on the qualitative properties and large time behavior for the following quasilinear equation involving a spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, $$ posed for $(x,t)\in\mathbf{R}^N\times(0,\infty)$, $N\geq1$, and in the range of exponents $1<m<p<\infty$, $σ>0$. We give a complete classification of (singular) self-similar solutions of the form $$ u(x,t)=t^{-α}f(|x|t^{-β}), \ α=\frac{σ+2}{σ(m-1)+2(p-1)}, \ β=\frac{p-m}{σ(m-1)+2(p-1)}, $$ showing that their form and behavior strongly depends on the critical exponent $$ p_F(σ)=m+\frac{σ+2}{N}. $$ For $p\geq p_F(σ)$, we prove that all self-similar solutions have a tail as $ξ\to\infty$ of one of the forms $$ u(x,t)\sim C|x|^{-(σ+2)/(p-m)} \quad {\rm or} \quad u(x,t)\sim \left(\frac{1}{p-1}\right)^{1/(p-1)}|x|^{-σ/(p-1)}, $$ while for $m<p<p_F(σ)$ we add to the previous the \emph{existence and uniqueness} of a \emph{compactly supported very singular solution}. These solutions will be employed in describing the large time behavior of general solutions in a forthcoming paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00349 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions Iagar, Razvan Gabriel Munteanu, Diana Rodica Analysis of PDEs This is the first of a two-parts work on the qualitative properties and large time behavior for the following quasilinear equation involving a spatially inhomogeneous absorption $$ \partial_tu=Δu^m-|x|^σu^p, $$ posed for $(x,t)\in\mathbf{R}^N\times(0,\infty)$, $N\geq1$, and in the range of exponents $1<m<p<\infty$, $σ>0$. We give a complete classification of (singular) self-similar solutions of the form $$ u(x,t)=t^{-α}f(|x|t^{-β}), \ α=\frac{σ+2}{σ(m-1)+2(p-1)}, \ β=\frac{p-m}{σ(m-1)+2(p-1)}, $$ showing that their form and behavior strongly depends on the critical exponent $$ p_F(σ)=m+\frac{σ+2}{N}. $$ For $p\geq p_F(σ)$, we prove that all self-similar solutions have a tail as $ξ\to\infty$ of one of the forms $$ u(x,t)\sim C|x|^{-(σ+2)/(p-m)} \quad {\rm or} \quad u(x,t)\sim \left(\frac{1}{p-1}\right)^{1/(p-1)}|x|^{-σ/(p-1)}, $$ while for $m<p<p_F(σ)$ we add to the previous the \emph{existence and uniqueness} of a \emph{compactly supported very singular solution}. These solutions will be employed in describing the large time behavior of general solutions in a forthcoming paper. |
| title | A porous medium equation with spatially inhomogeneous absorption. Part I: Self-similar solutions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2406.00349 |