Scattering for a quasilinear hyperbolic equation of Kirchhoff type
The global solvability and the existence of the scattering operator for the quasilinear hyperbolic Kirchhoff type equation \[ {\partial^2 u\over \partial t^2}= m\bigl( \| \nabla u\|^2 \bigr)^2 \Delta u\quad \text{in} \quad \mathbb{R}^n_x \times \mathbb{R}_t \tag{1} \] \[ u(x,0) =\varphi_0 (x),\quad {\partial u\over \partial t} (x,0)= \psi_0 (x) \tag{2} \] are studied, where \(m\in C^1 ([0,\infty)]\) and \(\inf_{\lambda\geq 0} m(\lambda) =m_0>0\). The wave and scattering operators in a neighborhood of the origin in some suitable function space are constructed, and it is shown the continuity of the scattering operator with respect to a suitable topology. Global solvability is proven using an a priori estimate for the local solutions of (1), (2). The asymptotic behavior of the Cauchy problem is also analysed, and an extension of local solvability is given for the abstract form.
scientific article; zbMATH DE number 1130533
scientific article