Asymptotic behavior of radially symmetric solutions for a quasilinear hyperbolic fluid model in higher dimensions

• Published in: Journal of Differential Equations Vol. 262; no. 10; pp. 5133 - 5159
• Main Authors: Hashimoto, Itsuko, Kozono, Hideo
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.05.2017
• Subjects: Asymptotic behavior; Damped wave equation; Galerkin method; Stationary wave; secondary; Damped wave equation; Galerkin method; Stationary wave; Asymptotic behavior; primary
• ISSN: 0022-0396; 1090-2732
• DOI: 10.1016/j.jde.2017.01.020

We consider the large time behavior of the radially symmetric solution to the equation for a quasilinear hyperbolic model in the exterior domain of a ball in general space dimensions. In the previous paper [2], we proved the asymptotic stability of the stationary wave of the Burgers equations in the same exterior domain when the solution is also radially symmetric. On the other hand, in the 1D-case, a similar asymptotic structure as above to the damped wave equation with a convection term has been established by Ueda [10] and Ueda–Kawashima [11]. Assuming a certain condition on the boundary data on the ball and the behavior at infinity of the fluid, we shall prove that the stationary wave of our quasilinear hyperbolic model is asymptotically stable. The weighted L2-energy method plays a crucial role in removing such a restriction on the sub-characteristic condition on the stationary wave.