Communications in Mathematical Sciences

Volume 19 (2021)

Number 3

Global existence of solution in the Besov space to the nonlinear wave equations in $\mathbb{R}^d$

Pages: 629 – 646

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n3.a3

Authors

Xinglong Wu (Center for Mathematical Sciences and Department of Mathematics, Wuhan University of Technology, Wuhan, China)

Boling Guo (Institute of Applied Physics and Computational Mathematics, Beijing, China)

Abstract

In [T.C. Sideris, Comm. Part. Diff. Eqs., 8:1291–1323, 1983], the author proves the solution of the nonlinear wave equations breaks down in finite time, if the initial data is radially symmetric and arbitrarily small. The present article is devoted to the study of the lower bound of blow-up rate of blow-up solution and the global solution to a class of nonlinear wave equations in $\mathbb{R}^d , d \gt 3$. We first recall some useful lemmas in Besov spaces. Next, the local well-posedness of Equation (1.1) is obtained in $\dot{B}^{\frac{d}{2} - \frac{1}{2}}_{2,1} \cap \dot{B}^{\frac{d}{2}}_{2,1}$, and a lower bound of blow-up rate of blow-up solution in the space is established. Finally, by construction of the space $\mathscr{X}_R (M)$, thanks to the contraction mapping argument, we derive the global solution for the Cauchy problem of Equation (1.1) if the initial datum is sufficiently small.

Keywords

nonlinear wave equations, well-posedness, Besov spaces, Bony decomposition, lower bound of blow-up rate, global solution

2010 Mathematics Subject Classification

35G60, 35L05

Received 13 June 2020

Accepted 29 September 2020

Published 5 May 2021