Mixed problems for evolution equations
Reiko Sakamoto · Kyoto journal of mathematics · 1984
Researches for heat equations or wave equations have a long h is to ry .In 1938, Petrowski studied Cauchy problem for evolution equations a s a generalization of above equations.Moreover, he specialized two essential types of evolution equations, i.e. p-parabolic equations and strictly hyperbolic equations ([1]).After him, many authors studied the two types of equations in separate w ays.O n the other hand, recently, Volevich-Gindikin gave a concept of dominantly correct evolution equations, which are 1-1'-well posed under any change of lower order terms in the sense of Newton polygon ([2]).In this paper, we shall show the'll'-weil posedn'es § of mixed problems for domi- nantly correct evolution equations, assuming th e uniform Lopatinski conditions.The process of the analysis of our problem is just pararel to that in [3].Our problem is to seek a solution u satisfyingwhere ff, g 1 , t i 4 are given datas and Q is a domain in R " (we only deal with the case when 52= R _O .Our main result in this paper is T heorem Under the assum ptions (A), (B) and (B*), the problem (P ) is H "well posed.8 1 .Newton polygon 1.1.Newton polygon.For a polynomial A(T, E a " .r a y , eT,1, we define the Newton polygon of A by NA=convex hull of tl A u where