Decay Estimates for Fourth Order Wave Equations

may be thought of as a nonlinear beam equation. In this paper we obtain both L-L estimates and space-time integrability estimates on solutions to the linear equation. We also use these estimates to study the local existence and asymptotic behavior of solutions to the nonlinear equation, for nonlinear terms which grow like a certain power of u. The main L-L estimate (Theorem 2.1) states that solutions of the linear equation with initial data (u(0), ut(0)) in W 2,q(Rn) ⊕ Lq(Rn) are bounded in L(R) ⊕ W−2,q(Rn) for 2 ≤ q ≤ 2∗∗ for all time and that their norm in this space decays at the (optimal) rate t n 2q−4 . Here and throughout 2∗∗ = { 2n n−4 for n ≥ 5 ∞ for 1 ≤ n ≤ 4 (1.2)