Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2021
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866929230946762752 |
|---|---|
| author | Iagar, Razvan Gabriel Muñoz, Ana Isabel Sánchez, Ariel |
| author_facet | Iagar, Razvan Gabriel Muñoz, Ana Isabel Sánchez, Ariel |
| contents | We prove existence and uniqueness of a global in time self-similar solution growing up as $t\to\infty$ for the following reaction-diffusion equation with a singular potential $$ u_t=Δu^m+|x|^σu^p, $$ posed in dimension $N\geq2$, with $m>1$, $σ\in(-2,0)$ and $1<p<1-σ(m-1)/2$. For the special case of dimension $N=1$, the same holds true for $σ\in(-1,0)$ and similar ranges for $m$ and $p$. The existence of this global solution prevents finite time blow-up even with $m>1$ and $p>1$, showing an interesting effect induced by the singular potential $|x|^σ$. This result is also applied to reaction-diffusion equations with general potentials $V(x)$ to prevent finite time blow-up via comparison. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_04806 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential Iagar, Razvan Gabriel Muñoz, Ana Isabel Sánchez, Ariel Analysis of PDEs We prove existence and uniqueness of a global in time self-similar solution growing up as $t\to\infty$ for the following reaction-diffusion equation with a singular potential $$ u_t=Δu^m+|x|^σu^p, $$ posed in dimension $N\geq2$, with $m>1$, $σ\in(-2,0)$ and $1<p<1-σ(m-1)/2$. For the special case of dimension $N=1$, the same holds true for $σ\in(-1,0)$ and similar ranges for $m$ and $p$. The existence of this global solution prevents finite time blow-up even with $m>1$ and $p>1$, showing an interesting effect induced by the singular potential $|x|^σ$. This result is also applied to reaction-diffusion equations with general potentials $V(x)$ to prevent finite time blow-up via comparison. |
| title | Self-similar solutions preventing finite time blow-up for reaction-diffusion equations with singular potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2111.04806 |