Summability of formal solution of Cauchy problem for some PDE with variable coefficients (Recent development of micro-local analysis for the theory of asymptotic analysis)
Year: 2013
Type: article
Abstract: We give a proof of the summability of formal solution for Cauchy problem of linear first order partial differential equations with respect to t and with t dependent coefficients under some global conditions for Cauchy data. §1. ResultWe consider the following Cauchy problem for linear partial differential equations of first order with respect to t and with t dependent coefficients (1.1) \ l e f t \ { \ b e g i n { a r r a y } { l } @ u ( t ; x ) = \ s u m _ { ( i , $ \ a l p h a $ ) } { f { i n i t e } } a _ { i $ \ a l p h a $ } t { i } \ p a r t i a l _ { x } { $ \ a l p h a $ } u ( t , x ) \ \ u ( 0 , x ) = $ \ v a r p h i $ ( x ) \ i n \ m a t h c a l { O } , \ e n d { a r r a y } \ r i g h t .where (t, x)\in \mathbb{C}^{2}, a_{ i $\al pha$} \i n \mat hbb{ C} and \ma t hc a l { O} denotes the set of holomorphic functions in a neigh- borhood of the origin.The Cauchy problem (1.1) has a unique formal solution of the form (1.2) û ( t , x ) = \ d i s p l a y s t y l e \ s u m_ { n \ g e q 0 } u _ { n } ( x ) \ f r a c { t {n } } { n !} .We have an interest in the case where the formal solution is divergent.We assume that for the operator (1.3) \displaystyle \max $\alpha$>1,
Author: Kunio Ichinobe
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W598636593