The optimal large time behavior of 3D quasilinear hyperbolic equations with nonlinear damping

We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping. The main novelty of this paper is two-fold. First, we prove the optimal decay rates of the second and third order spatial derivatives of the solution, which are the same as those of the heat equation, and in particular, are faster than ones of previous related works. Second, for well-chosen initial data, we also show that the lower optimal L² convergence rate of the k (∈ [0, 3])-order spatial derivatives of the solution is \({(1 + t)^{ - {{3 + 2k} \over 4}}}\) $${(1+t)}^{-\frac{3+2k}{4}}$$ . Therefore, our decay rates are optimal in this sense. The proofs are based on the Fourier splitting method, low-frequency and high-frequency decomposition, and delicate energy estimates.