Asymptotic Behavior of Solutions of a Degenerate Diffusion Equation with a Multistable Reaction
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909655211442176 |
|---|---|
| author | Li, Fang Lou, Bendong |
| author_facet | Li, Fang Lou, Bendong |
| contents | We consider a generalized degenerate diffusion equation with a reaction term $u_t=[A(u)]_{xx}+f(u)$, where $A$ is a smooth function satisfying $A(0)=A'(0)=0$ and $A(u),\ A'(u),\ A''(u)>0$ for $u>0$, $f$ is of monostable type in $[0,s_1]$ and of bistable type in $[s_1,1]$. We first give a trichotomy result on the asymptotic behavior of the solutions starting at compactly supported initial data, which says that, as $t\to \infty$, either small-spreading (which means $u$ tends to $s_1$), or transition, or big-spreading (which means $u$ tends to $1$) happens for a solution. Then we construct the classical and sharp traveling waves (a sharp wave means a wave having a free boundary which satisfies the Darcy's law) for the generalized degenerate diffusion equation, and then using them to characterize the spreading solution near its front. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_17177 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Behavior of Solutions of a Degenerate Diffusion Equation with a Multistable Reaction Li, Fang Lou, Bendong Analysis of PDEs 35K65, 35K57, 35B40, 35K15 We consider a generalized degenerate diffusion equation with a reaction term $u_t=[A(u)]_{xx}+f(u)$, where $A$ is a smooth function satisfying $A(0)=A'(0)=0$ and $A(u),\ A'(u),\ A''(u)>0$ for $u>0$, $f$ is of monostable type in $[0,s_1]$ and of bistable type in $[s_1,1]$. We first give a trichotomy result on the asymptotic behavior of the solutions starting at compactly supported initial data, which says that, as $t\to \infty$, either small-spreading (which means $u$ tends to $s_1$), or transition, or big-spreading (which means $u$ tends to $1$) happens for a solution. Then we construct the classical and sharp traveling waves (a sharp wave means a wave having a free boundary which satisfies the Darcy's law) for the generalized degenerate diffusion equation, and then using them to characterize the spreading solution near its front. |
| title | Asymptotic Behavior of Solutions of a Degenerate Diffusion Equation with a Multistable Reaction |
| topic | Analysis of PDEs 35K65, 35K57, 35B40, 35K15 |
| url | https://arxiv.org/abs/2501.17177 |