On existence and scattering with minimal regularity for semilinear wave equations. (English) Zbl 0846.35085

The authors study the semilinear Cauchy problem \[ (\partial^2_t- \Delta) u= F_\kappa(u),\quad u(0, x)= f(x)\in \dot H^\gamma(\mathbb{R}^n),\quad \partial_t u(0, x)= g(x)\in \dot H^{\gamma- 1}(\mathbb{R}^n), \] where \(u\geq 2\), \(\kappa> 1\) and \(F_\kappa\) is assumed to be a \(C^1\) function satisfying \(|F_\kappa(u)|\leq C|u|^\kappa\), \(C^{- 1}|F_\kappa(u)|\leq |u F_\kappa'(u)|\leq C|F_\kappa(u)|\). Here \(\dot H^\gamma(\mathbb{R}^n)\) is the homogeneous Sobolev space with norm \(|f|_{\dot H^\gamma}= ||D_x|^\gamma f|_{L^2}\), \(|D_x|= \sqrt{- \Delta_x}\).
The main goal of the paper is to find the minimal \(\gamma\), depending on \(\kappa\) and \(n\), such that the initial value problem has a weak local solution. The authors study this problem in great generality and give sharp results. One of their results is that there is local existence, provided that \[ \kappa< \kappa(\gamma)= \begin{cases} 1+ {4\over (n+ 1)- 4\gamma},\quad & \text{if}\quad \gamma_0< \gamma\leq 1/2\\ 1+ {4\over n- 2\gamma},\quad & \text{if} \quad 1/2\leq \gamma< n/2,\end{cases} \] with \(\gamma_0=(n- 3)/2(n- 1)\), \(n\geq 3\), and \(\gamma_0= 1/4\) for \(n= 2\). The paper also contains uniqueness results and results on global existence and existence of the scattering operator for small data. The proofs use Strichartz-type estimates.


35L70 Second-order nonlinear hyperbolic equations
35P25 Scattering theory for PDEs
35L15 Initial value problems for second-order hyperbolic equations
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