Convergence to pushed fronts and the behavior of level sets in monostable reaction-diffusion equations
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912894487101440 |
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| author | Kiyono, Ryo |
| author_facet | Kiyono, Ryo |
| contents | We study the behavior of solutions of a monostable reaction-diffusion equation $u_t=Δ_x u +u_{yy} +f(u)$ ($x \in \mathbb{R}^{n-1}$, $y \in \mathbb{R}$, $t>0$), with the unstable equilibrium point $0$ and the stable equilibrium point $1$. Under the condition that the corresponding one-dimensional equation has a pushed front $Φ_{c^*}(z)$ with $Φ_{c^*}(-\infty)=1$, $Φ_{c^*}(\infty)=0$, we show that the solution $u(x,y,t)$ approaches $Φ_{c^*}(y-γ(x,t))$ for some $γ(x,t)$ as $t \to \infty$, if initially $u(x,y,0)$ decays sufficiently fast as $y \to \infty$ and is bounded below by some positive constant near $y=-\infty$. It is also shown that $γ(x,t)$ is approximated by the mean curvature flow with a drift term. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_09806 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence to pushed fronts and the behavior of level sets in monostable reaction-diffusion equations Kiyono, Ryo Analysis of PDEs 35K57, 35B40 We study the behavior of solutions of a monostable reaction-diffusion equation $u_t=Δ_x u +u_{yy} +f(u)$ ($x \in \mathbb{R}^{n-1}$, $y \in \mathbb{R}$, $t>0$), with the unstable equilibrium point $0$ and the stable equilibrium point $1$. Under the condition that the corresponding one-dimensional equation has a pushed front $Φ_{c^*}(z)$ with $Φ_{c^*}(-\infty)=1$, $Φ_{c^*}(\infty)=0$, we show that the solution $u(x,y,t)$ approaches $Φ_{c^*}(y-γ(x,t))$ for some $γ(x,t)$ as $t \to \infty$, if initially $u(x,y,0)$ decays sufficiently fast as $y \to \infty$ and is bounded below by some positive constant near $y=-\infty$. It is also shown that $γ(x,t)$ is approximated by the mean curvature flow with a drift term. |
| title | Convergence to pushed fronts and the behavior of level sets in monostable reaction-diffusion equations |
| topic | Analysis of PDEs 35K57, 35B40 |
| url | https://arxiv.org/abs/2602.09806 |