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Global solutions in the critical Sobolev space for the wave equations with nonlinearity of exponential growth. (English) Zbl 0931.35107

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 \(\dot H^{1/2} \times\dot H^{-1/2}\).

MSC:

35L70 Second-order nonlinear hyperbolic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
35L15 Initial value problems for second-order hyperbolic equations
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