A porous medium equation with dominating weighted absorption: three types of self-similar solutions
Abstract
Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption ∂tu=Δum−|x|σup, (x,t)∈realN×(0,∞), N≥1, with exponents 1<p<m and σ>0, are classified. Looking for solutions in the form u(x,t)=t−αf(|x|tβ), α=(σ+2)/(σ(m−1)+2(p−1)), β=(m−p)/(σ(m−1)+2(p−1)), it is shown that all their profiles satisfy the behavior at infinity given by limξ→∞ξσ/(p−1)f(ξ)=(1/(p−1))1/(p−1), but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with f(0)>0, f'(0)=0, another unique solution such that f presents a \emph{dead-core}; that is, f≡0 for ξ∈[0,ξ0] for some ξ0>0, and, finally, there exists K*∈(0,∞) such that, for any K∈(0,K*), there is at least a solution such that limξ→0ξ−(σ+2)/(m−p)f(ξ)=K. The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.
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BibTeX
@misc{iagar2026porous,
title = {{A porous medium equation with dominating weighted absorption: three types of self-similar solutions}},
author = {Razvan Gabriel Iagar and Diana-Rodica Munteanu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20397},
eprint = {2609.20397},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20397},
}
RIS
TY - UNPB
TI - A porous medium equation with dominating weighted absorption: three types of self-similar solutions
AU - Razvan Gabriel Iagar
AU - Diana-Rodica Munteanu
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20397
UR - https://arxiv.org/abs/2609.20397
AB - Self-similar solutions to the porous medium equation with dominating spatially inhomogeneous absorption ∂_tu=Δu^m−|x|^σu^p, (x,t)∈real^N×(0,∞), N≥1, with exponents 1<p<m and σ>0, are classified. Looking for solutions in the form u(x,t)=t^−αf(|x|t^β), α=(σ+2)/(σ(m−1)+2(p−1)), β=(m−p)/(σ(m−1)+2(p−1)), it is shown that all their profiles satisfy the behavior at infinity given by lim_ξ→∞ξ^σ/(p−1)f(ξ)=(1/(p−1))^1/(p−1), but the solutions strongly differ with respect to their behavior near the origin: there exist a unique solution with f(0)>0, f'(0)=0, another unique solution such that f presents a \emph{dead-core}; that is, f≡0 for ξ∈[0,ξ_0] for some ξ_0>0, and, finally, there exists K^*∈(0,∞) such that, for any K∈(0,K^*), there is at least a solution such that lim_ξ→0ξ^−(σ+2)/(m−p)f(ξ)=K. The large time behavior of general solutions, making strong use of these three types of self-similar solutions, will be addressed in a companion work.
ER -