Fractional derivatives of composite functions and the Cauchy problem for the nonlinear half wave equation

2017/07/26 by Hidano, Kunio, Wang, Chengbo · 2 citations
#35F25 #35L15 #35L70 #42B25 #42B37 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1707.08319

Abstract

We show new results of wellposedness for the Cauchy problem for the half wave equation with power-type nonlinear terms. For the purpose, we propose two approaches on the basis of the contraction-mapping argument. One of them relies upon the L_t^q L_x^∞ Strichartz-type estimate together with the chain rule of fairly general fractional orders. This chain rule has a significance of its own. Furthermore, in addition to the weighted fractional chain rule established in Hidano, Jiang, Lee, and Wang (arXiv:1605.06748v1 [math.AP]), the other approach uses weighted space-time L² estimates for the inhomogeneous equation which are recovered from those for the second-order wave equation. In particular, by the latter approach we settle the problem left open in Bellazzini, Georgiev, and Visciglia (arXiv:1611.04823v1 [math.AP]) concerning the local wellposedness in H^s_{\rm rad}(\mathbb Rⁿ) with s>1/2.

Cited by

• Existence of solutions of semilinear wave equations with time-dependent propagation speed and time derivative nonlinearity
• Existence of solutions to the semilinear damped wave equation with non-L² slowly decaying data : polynomial nonlinearity case

Related

• Weighted fractional chain rule and nonlinear wave equations with minimal regularity
• Well-posedness results for a class of semi-linear super-diffusive\n equations
• Fractional derivative of composite functions: exact results and physical applications
• Local well-posedness of the higher order nonlinear Schrödinger equation on the half-line: single boundary condition case
• On the well-posedness of time-space fractional Schrödinger equation on ℝ^d