Global in time solutions of evolution equations in scales of Banach function spaces in

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• 作者：Daniel Gourdin ; Todor Gramchev
• 关键词：Cauchy problem ; Global solutions ; Gevrey spaces ; Picard iteration scheme ; Contraction principle ; Singular Gronwall estimates
• 刊名：Bulletin des Sciences Mathematiques
• 出版年：2007
• 出版时间：December 2007
• 年：2007
• 卷：131
• 期：8
• 页码：761-786
• 全文大小：298 K

文摘

We investigate the global well-posedness of the Cauchy problem for linear evolution partial differential equations d71d5e66c2b70db455f6e86e6060"" title=""Click to view the MathML source"">P(t,x,∂_t,∂_x)u=f(t,x) with coefficients depending on , , or , unbounded for |x|→+∞. We introduce the notion of a critical Gevrey-d75e28afaa2d474ae9a18621167"" title=""Click to view the MathML source"">C^∞ index 0<σ_cr+∞ for P. Typically, the coefficients are supposed to be analytic-Gevrey regular in x while with respect to t they are holomorphic (respectively, continuous) if (respectively, ). Coefficients with singularity at t=0 of the type O(|t|^−ρ), d7c2c31"" title=""Click to view the MathML source"">0<ρ<1, are also considered when . A description of the Gevrey critical index is given by means of Newton polyhedra geometry. We propose a unified novel approach, based on deriving convergence of parameter depending Picard successive approximations, provided contraction perturbed with singular Gronwall type estimates hold. The crucial ingredient consists of a suitable choice of multi-parameter scales of Banach function spaces and detailed analysis of integral equations in such spaces. The outcome is a series of new Cauchy–Kovalevskaya–Nagumo type theorems for global in time well-posedness of the Cauchy problem in for both inductive and projective Gevrey spaces of index σσ_cr, as well as in the C^∞ class if σ_cr=+∞, provided the coefficients of the σ-“dominating part” of P are polynomials obeying certain conditions which turn out to be sharp for space dimension n=1. For n2 we show new global Cauchy–Kovalevskaya–Nagumo type theorems allowing arbitrary growth with respect to x of some coefficients for class of operators provided they obey some global reduction to Poincaré type normal form. We recapture as particular cases the main results of [D. Gourdin, M. Mechab, Solutions globales d'un problème de Cauchy linéaire, J. Funct. Anal. 202 (1) (2003) 123–146] in projective Gevrey spaces for Kovalevskaya and hyperbolic equations under weaker restrictions.