Variational Approaches to Three-Phase Standing and Traveling Waves of Reaction-Diffusion-Gradient Systems with a Triple-Well Potential

• Main Authors: Hung-Yu Chien, 簡鴻宇
• Other Authors: 陳俊全
• Format: Others
• Language: en_US
• Published: 2019
• Online Access: http://ndltd.ncl.edu.tw/handle/5x7h4w

博士 === 國立臺灣大學 === 數學研究所 === 107 === In this paper we aim to find standing and traveling wave solutions, i.e. w(z, y) = u(x, y, t) with z = x − ct, to the reaction-diffusion gradient system with a triple- well potential ∂tu = ∆u − ∇W(u) on an entire domain R2 or a cylindrical domain R×(−l,l). Firstly by the theory of Γ-convergence, standing wave solutions (i.e. stationary solutions) are obtained under a condition that the potential W is invariant under a simple reflection. This symmetry assumption is weaker than the invariance under a general symmetric group, which is assumed by some literatures. And also, under the same condition on symmetry, via a variational method, we can show the existence of a traveling wave solution that connects the three constant equilibria in an approximate sense on a cylindrical domain. We propose a convexity condition on one of the equilibria of W to ensure the asymptotical convergence to this equilibrium of the traveling wave solutions at z = −∞.