Global Cauchy problems on \(\mathbb{R}^n\) for weakly hyperbolic systems with coefficients admitting superlinear growth for \(| x| \rightarrow \infty\)
The authors study the Cauchy problem for the hyperbolic system of pseudodifferential equations of the form \[ \begin{cases} h_1(t,x,D_t,D_x) u+h_0(t,x,D_x) u=f(t,x),\quad t\in \mathbb{R}, \quad x\in\mathbb{R}^n,\\ u(0,x)=u_0(x),\quad x\in \mathbb{R}^n. \end{cases}\tag{1} \] It is assumed that \(h_1=I_m D_t-a(t,x,D_x)\) is an \(m\times m\) matrix-valued first order partial pseudodifferential operator in \(t\), \(h_0=b(t,x,D_x)\) is an \(m\times m\) matrix-valued zero order partial pseudodifferential operator in \(x\), smoothly depending on \(t\), \(D_x=i^{-1}\partial_x\). Under some conditions on the coefficients of principal symbol \(H(t,x,\tau,\xi)=\tau I_v-a(t,x,\xi)\), which allow the superlinear growth for \(|x|\to\infty\) and simultaneously double characteristic root, the authors prove the globally well-posedness of the problem (1) in some scale of two-indexed weighted Sobolev spaces \[ CH^{s,W}(\mathbb{R}^n)=\{u\in C([0,T];S'(\mathbb{R}^n);e^{-W(t,\langle x\rangle)}u(t,x)\in C([0,T];H^s(\mathbb{R}^n))\} \] with norm \(\|u\|_{T;W,s}=\sup_{t\in [0,T]}\|e^{-W(t,\langle\cdot\rangle)}\langle\cdot\rangle^{s_2} \langle D\rangle^{s_1}u(t,\cdot)\|_{L^2}\) (here \(\langle x\rangle =(1+|x|^2)^{1/2}\), and \(\langle D\rangle\) denotes the Fourier multiplier with symbol \(\langle\xi\rangle\)). Also, they establish \textit{a priori} estimates for solutions to the problem (1) in the norm of \(CH^{s,W}(\mathbb{R}^n)\).
scientific article; zbMATH DE number 7005794