Some Remarks on the Well-Posedness of the Cauchy Problem in Gevrey Spaces

Abstract

In [2] ⁽¹⁾ F.Colombini, E.De Giorgi, S.Spagnolo proved that the Cauchy problem

$$ \left\{ {\matrix{ {\partial _{tt}^2u{\rm{ - }}\sum\nolimits_{i,j = 1 }^n {{c_{ij}}\left( t \right)\partial _{xixj}^2u = 0} } & {\left( {t,x} \right){\rm{ }} \in {\rm{]0,}}T[ \times {R^n}} \cr {u\left( {0,x} \right){\rm{ = }}{u_0}\left( x \right)} & {x \in {R^n}} \cr {{\partial _t}u\left( {0,x} \right){\rm{ = }}{u_1}\left( x \right)} & {x \in {R^n}} \cr } } \right. $$

((1.1))

where c _ij are real integrable functions on [0,T] such that for some constant a ₀ > 0

$$\sum\limits_{i,j = 1}^n {{c_{ij}}\left( t \right)} {\xi _i}{\xi _j}{\rm{ }} \ge {\rm{ }}{{\rm{a}}_0}|\xi {|^2}{\rm{ , }}\xi \in {{\rm{R}}^n},$$

is well posed in the Gevrey space ε ^{3}(R ⁿ_x ), for s ∊ [1,1/(1 — X)[, X ^∊]0,1[, if there exists c ≥ 0 such that

$$\int_O^T {|{c_{ij}}} (t + \tau ) - {c_{ij}}(t)|dt {\rm{ < }}C{\tau ^x},{\rm{ }}\tau {\rm{ > }}0{{\rm{ }}^{{\rm{(2)}}}}{\rm{.}}$$

((1.2))