Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth
The authors show the existence of global solutions of the initial value problem \[ \partial^2_t u(t,x)- \Delta u(t,x)= f\bigl(u(t,x) \bigr) \] \[ u(0,\cdot) =\varphi\in \dot H^{n/2} (\mathbb{R}^n), \] \[ \partial_t u(0, \cdot)= \psi\in \dot H^{n/2-1} (\mathbb{R}^n), \] where \(f\) is a function belonging to some class of functions whose typical example is \(u^2e^{\lambda|u|^2}\), \(\lambda\geq 0\). It is pointed out that for various reasons \(n/2\) is considered to be a critical value. Let \[ Y=(\dot H^{n/2}\times\dot H^{n/2-1}) \cup(\dot H^{1/2} \times\dot H^{-1/2}), \] \[ K(t)\equiv (\sin t\sqrt{- \Delta})/ \sqrt{-\Delta}, \quad\dot K(t)\equiv \cos t\sqrt {-\Delta}. \] It is shown that if \((\varphi, \psi)\in Y\) with small norm, the above problem has a unique solution which behaves like \(\dot K(t) \varphi_+ +K(t)\dot \psi_+\) in the norm of \(Y\) as \(t\to +\infty\) for some \((\varphi_+,\psi_+)\in Y\). Furthermore, for \((\varphi_-, \psi_-) \in Y\) with sufficiently small norm there exists a solution which behaves like \(\dot K(t) \varphi_- +K(t)\psi_-\) as \(t\to-\infty\). The scattering operator \((\varphi_-, \psi_-)\to (\varphi_+, \psi_+)\) is continuous in the norm of \(\dot H^{1/2} \times\dot H^{-1/2}\).
scientific article; zbMATH DE number 1336879