Ruipeng Shen — Page 3

arXiv2012

I-method for Defocusing, Energy-Subcritical Non-linear Wave Equation

In this paper the author considers the global existence and well-posedness of the non-linear wave equation $\partial_t^2 u - Δu = -|u|^{p-1} u$ in 3-dimensional space, assuming that the initial data is in the space $(\dot{H}^s \cap \dot{H}^{s_p}) \times (\dot{H}^{s-1} \cap \dot{H}^{s_p-1})$, with $11/3<p<5$, $s_p = 3/2 - 2/(p-1)$ and $s$ sufficiently close to 1. The main idea is to use the I-method and consider the almost conservation law of the energy.

math.AP↗

arXiv2011

Global Well-posedness and Scattering of Defocusing Energy subcritical Nonlinear Wave Equation in dimension 3 with radial data

The paper deals with the defocusing case of the energy subcritical non-linear wave equation in $R^3$. We assume the initial data is in the space $\dot{H}^s \times \dot{H}^{s-1}$ and radial. If $s=1$, this is the energy space and the scattering results are known. In this paper we choose $15/16<s<1$ and assume that the nonlinearity equals $-|u|^p u$ with $p = 2/{3/2 -s}$, so that this problem is $\dot{H}^s \times \dot{H}^{s-1}$ critical. We will prove the global well-posedness and scattering of the solution under the additional assumption that the $\dot{H}^s \times \dot{H}^{s-1}$ norm of the solution is uniformly bounded for all time $t$ in the maximal lifespan of the solution.

math.AP↗