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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Accesso online: | https://arxiv.org/abs/2404.11693 |
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| _version_ | 1866929318204014592 |
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| author | Alves, Claudianor O. Isneri, Renan J. S. Montecchiari, Piero |
| author_facet | Alves, Claudianor O. Isneri, Renan J. S. Montecchiari, Piero |
| contents | In this paper, we prove the existence, uniqueness and qualitative properties of heteroclinic solution for a class of autonomous quasilinear ordinary differential equations of the Allen-Cahn type given by
$$
-\left(ϕ(|u'|)u'\right)'+V'(u)=0~~\text{ in }~~\mathbb{R},
$$
where $V$ is a double-well potential with minima at $t=\pmα$ and $ϕ:(0,+\infty)\to(0,+\infty)$ is a $C^1$ function satisfying some technical assumptions. Our results include the classic case $ϕ(t)=t^{p-2}$, which is related to the celebrated $p$-Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case $ϕ(t)=\frac{1}{\sqrt{1+t^2}}$, which is directly associated with the prescribed mean curvature operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11693 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Uniqueness of Heteroclinic Solutions in a Class of Autonomous Quasilinear ODE Problems Alves, Claudianor O. Isneri, Renan J. S. Montecchiari, Piero Analysis of PDEs In this paper, we prove the existence, uniqueness and qualitative properties of heteroclinic solution for a class of autonomous quasilinear ordinary differential equations of the Allen-Cahn type given by $$ -\left(ϕ(|u'|)u'\right)'+V'(u)=0~~\text{ in }~~\mathbb{R}, $$ where $V$ is a double-well potential with minima at $t=\pmα$ and $ϕ:(0,+\infty)\to(0,+\infty)$ is a $C^1$ function satisfying some technical assumptions. Our results include the classic case $ϕ(t)=t^{p-2}$, which is related to the celebrated $p$-Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case $ϕ(t)=\frac{1}{\sqrt{1+t^2}}$, which is directly associated with the prescribed mean curvature operator. |
| title | Uniqueness of Heteroclinic Solutions in a Class of Autonomous Quasilinear ODE Problems |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.11693 |