Microlocal regularity for global solutions of partial differential equations with polynomial coefficients
The author studies regularity questions for solutions \(u\in E^ s({\mathbb{R}}^ n\times I)\) of differential equations \(p(x,t,D_ x,D_ t)u=0\), where I is an open interval in \({\mathbb{R}}\) with \(0\in I\), \(s>1\) is a rational number, \(E^ s({\mathbb{R}}^ n\times I)\) is the space of functions \(f\in C^{\infty}({\mathbb{R}}^ n\times I)\) for which growth type restrictions \(| D_ x^{\alpha}D^ j_ tf(x,t)| \leq c_{\alpha j} \exp (A| x|^ s)\) hold, and \(p(x,t,D_ x,D_ t)\) is of the form \[ p(x,t,D_ x,D_ t)=\sum_{| \alpha | /s+| \beta | /\sigma +j\leq m}a_{\alpha \beta j}(t)x^{\alpha}D_ x^{\beta}D^ j_ t \] with \(s^{- 1}+\sigma^{-1}=1\), \(a_{00m}(t)\equiv 1\) and \(a_{\alpha \beta j}\) real-analytic in I. The statement of the main result involves an analytic wave front set adapted to the study of problems in which growth restritions at infinity appear. A discussion of this wave front set already has been given in an earlier paper of the author [Astérisque 89/90, 163-203 (1981; Zbl 0498.35027)]. Also the main theorem of the present paper has been announced there without proof.
scientific article; zbMATH DE number 4059920
scientific article