On a characteristic Cauchy problem (Q1098994)

The author considers the characteristic Cauchy problem (C) \[ u_{ts}+\sum^{n}_{j,k=1}a_{jk} u_{x_ jx_ k}+\sum^{n}_{j=1}b_ j u\quad_{x_ j}+cu=f\quad in\quad [0,T]\times {\mathbb{R}}\times {\mathbb{R}}\quad n, \] \[ u(0,s,x)=g(s,x),\quad (s,x)\in {\mathbb{R}}\times {\mathbb{R}}\quad n. \] It is assumed that \(a_{jk},b_ j,c\in C^{\infty}([0,T]\times {\mathbb{R}}\times {\mathbb{R}}^ n\)), that the coefficients are constant outside of a compact subset of [0,T]\(\times {\mathbb{R}}\times {\mathbb{R}}^ n \)and that the matrix \((a_{jk})\) is negative definite. (If \((a_{jk})\) is positive definite, the change of variables \(s'=-s\) can be used.) Working with suitable Sobolev spaces \(H\) \(r_{\beta}\) with weight, (i.e. \(\| \phi \|_{H\quad r_{\beta}}=\| \exp (\beta s)\phi \|_{H\quad r})\), the author proves an existence and uniqueness theorem: If \(\beta <0\), \(f\in L\) \(2([0,T];H\) \(r_{\beta})\), \(g\in H_{\beta}^{r+1}\), then there exists a unique \(u\in C\) \(0([0,T];H_{\beta}^{r+1})\), which is a solution of (C). If \(f\in \cap^{m}_{k=0}C\) \(k([0,T];H_{\beta}^{r-k})\), then \(u\in \cap^{m}_{k=0}C\quad k([0,T];H_{\beta}^{r+1-k})\cap C^{m+1}(\quad [0,T];H_{\beta}^{r-1-m}).\) The range of influence is studied too.