arXiv subjects

Fan Dashan

Publications and source records attributed to Fan Dashan.

2 recordsLinked to original sources

arXiv2014

Cauchy problem for NLKG in modulation spaces with noninteger powers

In this paper, we consider the Cauchy problem for the nonlinear Klein-Gordon equation whose nonlinearity is $|u|^{k}u$ in the modulation space, where $k$ is not an integer. Our method can be applied to other equations whose nonlinear parts have regularity estimates. We also study the global solution with small initial value for the Klein-Gordon-Hartree equation. By this we can show some advantages of modulation spaces both in high and low regularity cases.

math.AP↗

arXiv2014

Critical exponent for evolution equation in Modulation space

In this paper, we propose a method to find the critical exponent for certain evolution equations in modulation spaces. We define an index $σ(s,q)$, and use it to determine the critical exponent of the fractional heat equation as an example. We prove that when $σ(s,q)$ is greater than the critical exponent, this equation is locally well posed in the space $C(0,T;M_{p,q}^{s})$; and when $σ(s,q)$ is less than the critical exponent, this equation is ill-posed in the space $C(0,T;M_{2,q}^{s})$. Our method may further be applied to some other evolution equations.

math.AP↗