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
Matthias Winter · Mathematical Methods in the Applied Sciences · 1995
Abstract We consider the equation (−1)m∇m (p∇mu) + ∂ u = ƒ in ℝn × (0, ∞) for arbitrary positive integers m and n and under the assumptions p − 1, ƒ ϵ C (ℝn) and p > 0. Even if the differential operator (−1)m∇m (p∇mu) has no eigenvalues, the solution u(x,t) may increase as t → ∞ for 2m ≥ 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∇mu) = 0, which satisfy appropriate asymptotic conditions at infinity. We also give an asymptotic characterization of the static limit.