Gevrey Asymptotic Representation of the Solutions of Equations with One Turning Point

Abstract

An ordinary differential equation of the type D T u + aj,,(t)taD{u = 0 ~ j+lallm,j<m with parameter ( E Rn and smooth coefficients ~j ,~ E C m ( [ -T , T ] ) is studied. It is assumed that all the characteristic roots of the equation vanish at t = 0 while for t # 0 they are real and distinct. The constructions of real-valued phase functions @kl ( k , 6 = 1, . . . ,rn) and of amplitude functions Ajkl such that for a given s E [-T,T] every solution u ( t , ( ) of the equation can be represented as m-1 m j = O l , k = l where \ k j ( s , ( ) = D:u(s, c), j = 0 , . . . , rn -1, are given. T > 0, Dt = -id/& a = (a1,a~, . . . ,an) is a multi-index, Icy1 = a1 +a2+. . . +a,. A point to E J is said to be a turning point of the equation (1.1) if there is a ( # 0 such that all roots A l ( t , () (1 = 1, . , . ,m) of the characteristic equation 1991 Mathematics Subject Classification. Primary 34 A99; Secondary 34 E 20, 35 L 30.

A generalized Levi condition for weakly hyperbolic Cauchy problems with coefficients low regular in time and smooth in space

“…A more general approach to Levi conditions is the use of shape functions, which describe the speed at which characteristics coincide. We introduce shape functions following [7,17,18]. Definition 1.3 (Shape functions).…”

“…Instead of a typical Gevrey Levi condition (see e.g. [17,18]), where we would assume that the coefficients of the lower order terms satisfy…”

Exponential Function of Pseudo-Differential Operators in Gevrey Spaces

“…More precisely, it suggests that the fundamental solution can be written as a product of the Fourier integral operator with zero-order elliptic symbol and the pseudo-differential operator. If k ≥ l − 1, then the last pseudo-differential operator is of finite order [15], while for l, k, and θ satisfying the above mentioned inequalities, it is a pseudo-differential operator of infinite order [11,16]. This operator is one of the main factors of the fundamental solution constructed in [16] as well as an important component of the proof of the uniqueness theorem for the degenerate elliptic operators [1].…”

“…If k ≥ l − 1, then the last pseudo-differential operator is of finite order [15], while for l, k, and θ satisfying the above mentioned inequalities, it is a pseudo-differential operator of infinite order [11,16]. This operator is one of the main factors of the fundamental solution constructed in [16] as well as an important component of the proof of the uniqueness theorem for the degenerate elliptic operators [1]. It turns out that this operator is an exponential function of some other pseudo-differential operator appearing after the so-called perfect diagonalization and an application of the Egorov's theorem.…”

“…Thus, the main ingredient of such construction is the exponential function of the pseudo-differential operator. The case of the operators with x-independent coefficients, which is applicable to the Gevrey classes of functions, is given in Section 5 [16]. In that article the Cauchy problem by means of the Fourier transform is reduced to the ordinary differential equation with the large parameter and a single turning point.…”