Asymptotically self-similar solutions of the damped wave equation

作者：Hamza M^*

来源：Nonlinear Analysis-Theory Methods & Applications, 2010, 73(9): 2897-2916.

DOI：10.1016/j.na.2010.06.047

• Damped wave equation
• Self-similar variables
• Asymptotic behavior
• Self-similar solutions

We consider the damped hyperbolic equation
epsilon u(tt) + u(t) = (a(xi)u xi)xi - vertical bar u vertical bar(p-1)u, (xi, tau) is an element of R x R(+), (1)
where epsilon > 0, a(xi) -> 1 as vertical bar xi vertical bar -> +infinity and 1 < p < 3.
We prove in this article that the exact self-similar solutions of the semi-linear parabolic equation obtained by setting epsilon = 0 and a(xi) equivalent to 1 in (1) are also asymptotically stable self-similar solutions of the Eq. (1). The proof of our result relies on various energy estimates rewritten in the variables xi / root tau, ln tau.

• 出版日期2010-11-1