Asymptotic Behavior and Regularity for Nonlinear Dissipative Wave Equations in R

We discuss several results on asymptotic behavior and regularity for wave equations with nonlinear damping utt − ∆u + |ut|m−1ut = 0 in R+ × Rn, where m > 1. In particular, we show that ∇u ∈ Lm+1(I × Rn) for all initial data (u, ut)|t=0 ∈ H1(Rn) × L2(Rn) when m ≤ (n + 2)/n and I ⊂ R+ is a compact interval. We also consider global well-posedness in Sobolev spaces Hk(Rn)×Hk−1(Rn) for k > 2. The strength of nonlinear damping is critical when m = n/(n − 2) and n ≥ 3, which makes classical techniques ineffective. We outline the proof of global well-posedness for m = 3 and n = 3 (critical) under the additional condition that (u, ut)|t=0 are spherically symmetric. Finally, we present scattering results for spherically symmetric global solutions.