Uniqueness of Heteroclinic Solutions in a Class of Autonomous Quasilinear ODE Problems
Abstract
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 AllenCahn type given by $$ \left(\phi(u')u'\right)'+V'(u)=0~~\text{ in }~~\mathbb{R}, $$ where $V$ is a doublewell potential with minima at $t=\pm\alpha$ and $\phi:(0,+\infty)\to(0,+\infty)$ is a $C^1$ function satisfying some technical assumptions. Our results include the classic case $\phi(t)=t^{p2}$, which is related to the celebrated $p$Laplacian operator, presenting the explicit solution in this specific scenario. Moreover, we also study the case $\phi(t)=\frac{1}{\sqrt{1+t^2}}$, which is directly associated with the prescribed mean curvature operator.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.11693
 arXiv:
 arXiv:2404.11693
 Bibcode:
 2024arXiv240411693A
 Keywords:

 Mathematics  Analysis of PDEs