Regularity and scattering for the wave equation with a critical nonlinear damping. (English) Zbl 1180.35363

The authors demonstrate that the nonlinear wave equation \(\square u+u_{t}^{3}=0\) is globally well-posed in radially symmetric Sobolev spaces \(H^{k}_{\text{rad}}(\mathbb R^{3})\times H^{k - 1}_{\text{rad}}(\mathbb R^{3})\) for all integers \(k>2\). This partially extends the well-posedness in \(H^{k}(\mathbb R^{3})\times H^{k - 1}(\mathbb R^{3})\) for all \(k\in \)[1,2]. Furthermore, they obtain the global existence of \(C^{\infty }\) solutions with radial \(C_{0}^{\infty }\) data. The regularity problem requires smoothing and non-concentration estimates in addition to the standard energy estimates, since the cubic damping is critical when \(k=2\). Besides, they establish scattering results for initial data \((u,u_{t})|_{t=0}\) in radially symmetric Sobolev spaces, as well.


35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
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