Uniqueness and asymptotic stability of time-periodic solution for the fractal Burgers equation

Abstract

The paper is concerned with the time-periodic (T-periodic) problem of the fractal Burgers equation with a T-periodic force on the real line. Based on the Galerkin approximates and Fourier series (transform) methods, we first prove the existence of T-periodic solution to a linearized version. Then, the existence and uniqueness of T-periodic solution to the nonlinear equation are established by the contraction mapping argument. Furthermore, we show that the unique T-periodic solution is asymptotically stable. This analysis, which is carried out in energy space $H^1(0,T;H^{\frac{\alpha }{2}}(R))\cap L^2(0,T;{\dot{H}}^{\alpha })$ with $1<\alpha <\frac{3}{2}$, extends the T-periodic viscid Burgers equation in \cite{5} to the T-periodic fractional case.

References in corpus (1)

• Global well-posedness of the critical Burgers equation in critical Besov spaces

Cited by in corpus (1)

• Uniqueness and stability of steady-state solution with finite energy to the fractal Burgers equation