Date of Completion
Traveling wave solution; Traveling speed; Allen–Cahn equation; Fractional Laplacian; Continuation method; Hamiltonian identity
Field of Study
Doctor of Philosophy
We show the existence of traveling wave solutions to the Allen-Cahn equation with fractional Laplacians. A key ingredient is the estimation of the traveling speed of traveling wave solutions. In the meantime, we prove some qualitative properties of the solution, e.g., monotonicity, polynomial decays at infinity, Hamiltonian identity and Modica type estimates, and non-degeneracy. Moreover, we prove that for any balanced bistable nonlinearity, the traveling speeds linearly depend on the perturbation parameters.
Zhao, Mingfeng, "Traveling Wave Solutions To The Allen-Cahn Equations With Fractional Laplacians" (2014). Doctoral Dissertations. 420.