Asymptotic characterization of standing waves and of the static limit for a class of wave equations of higher order with a variable coefficient and time-independent incitation

✍ Scribed by Matthias Winter

• Publisher: John Wiley and Sons
• Year: 1995
• Tongue: English
• Weight: 742 KB
• Volume: 18
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1670180205

✦ Synopsis

Abstract

We consider the equation (−1)^m^∇^m^ (p∇^m^u) + ∂u = ƒ in ℝ^n^ × (0, ∞) for arbitrary positive integers m and n and under the assumptions p − 1, ƒ ϵ C(ℝ^n^) and p > 0. Even if the differential operator (−1)^m^∇^m^ (p∇^m^u) has no eigenvalues, the solution u(x,t) may increase as t → ∞ for 2__m__ ≥ n. For this case, we derive necessary and sufficient conditions for the convergence of u(x,t) as t → ∞. Furthermore, we characterize the functions occurring in these conditions as solutions of the homogeneous static equation (−1)^m^∇^m^ (p∇^m^u) = 0, which satisfy appropriate asymptotic conditions at infinity. We also give an asymptotic characterization of the static limit.