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Cauchy problem in Gevrey classes for some evolution equations of Schrödinger type

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  • Mathematics


Agliardi, R. and Mari, D. Osaka J. Math. 40 (2003), 917–924 CAUCHY PROBLEM IN GEVREY CLASSES FOR SOME EVOLUTION EQUATIONS OF SCHR ¨ODINGER TYPE R. AGLIARDI and D. MARI (Received February 18, 2002) 1. Introduction In this paper the Cauchy problem in Gevrey classes is studied for some partial differential — or, more generally, pseudo-differential — equations of Schro¨dinger type, that is, for differential equations whose type of evolution is 2 and whose characteris- tic roots are real. Our aim is to determine some Gevrey index σ for which the well- posedness of the Cauchy problem holds in Gevrey classes of order σ. Such an index depends on the multiplicity of the characteristic roots and on the lower order terms. Our result was obtained in [2] in the special case of differential equations with con- stant leading coefficients. 2. Notation Let us first introduce some notation about Gevrey spaces. If σ ≥ 1, then γσ(R ) will denote the class of all the smooth functions such that: sup ∈R α∈N |∂α ( )| −|α|α!−σ < +∞ for some > 0. Now we define some Gevrey-Sobolev spaces (compare [4] and [5]). For ε > 0, σ ≥ 1, > 0, let Dσ ε2 (R ) denote the space of all functions such that ‖ ε〈 〉1/σ ‖ < +∞, where ‖ ‖ is the usual Sobolev norm in (R ). Note that, if ′ < and ε′ > ε, then Dσ ε2 (R ) ⊂ Dσ ε ′ ′ 2 (R ). In this paper the space of the functions belonging to Dσ ε 02 (R ) for some ε, will be denoted by Dσ2 (R ). Let ε( ) be a positive function of , ∈ [− ]. If ( ) ∈ Dσ ε( )2 (R ), for every ∈ [− ], let us denote ‖ ε( )〈 〉1/σ ( )‖ by ‖‖ ( )‖‖ε( ) σ . Let us now give some notation about pseudo-differential operators. We shall de- note by σ the class of the pseudo-differential operators ( ) whose symbol ( ξ) 918 R. AGLIARDI AND D. MARI satisfies the following condition: sup α β∈N sup ξ∈R |ξ|≥ |∂αξ β ( ξ)| 〈ξ〉|α|− −|α+β|α!−1β!−σ <∞ for some > 0, ≥ 0. Finally ( ξ) is called a σ-regularizing symbol if: sup β∈N sup ξ∈R |ξ|≥ | β ( ξ)| exp( 〈ξ〉1/σ) −|β|β!−σ <∞ for som

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