L1 Decay estimates for dissipative wave equations

✍ Scribed by Albert Milani; Yang Han

• Publisher: John Wiley and Sons
• Year: 2001
• Tongue: English
• Weight: 151 KB
• Volume: 24
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.222

✦ Synopsis

Abstract

Let u and v be, respectively, the solutions to the Cauchy problems for the dissipative wave equation
$$u_{tt}+u_t‐\Delta u=0$$\nopagenumbers\end
and the heat equation
$$v_t‐\Delta v=0$$\nopagenumbers\end

We show that, as $t\rightarrow+\infty$\nopagenumbers\end, the norms $|\partial_t^k,D_x^\alpha u(,\cdot,,t)|_{L^1({\rm R}^n)}$\nopagenumbers\end and $|\partial_t^k,D_x^\alpha v(,\cdot,,t)|_{L^1({\rm R}^n)}$\nopagenumbers\end decay to 0 with the same polynomial rate. This result, which is well known for decay rates in $L^p({\rm R}^n)$\nopagenumbers\end with $2\leq p\leq+\infty$\nopagenumbers\end, provides another illustration of the asymptotically parabolic nature of the hyperbolic equation (1). Copyright © 2001 John Wiley & Sons, Ltd.