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On the Cauchy problem in Banach scales with compact embeddings. (English. Russian original) Zbl 0878.34054

Sib. Math. J. 37, No. 5, 1028-1036 (1996); translation from Sib. Mat. Zh. 37, No. 5, 1167-1175 (1996).
The main result is the following statement: Let \(B_\rho\) \((0<\rho<\infty)\) be a \(K\)-scale of Banach spaces and let \(L_1,\dots,L_j\) be a finite collection of singular operators satisfying \[ |Lu|_{\rho_1}\leq{\omega\over \rho_2-\rho_1} |u|_{\rho_2},\quad\omega= \text{const}>0. \] Then there is an equivalent \(K\)-scale \(B_\rho'\) such that \(L=L_i\) \((i=1,\dots,j)\) satisfy the quasidifferential estimate \[ |Lu|_{\rho}\leq {\partial\over\partial\rho} |u|_{\rho}. \]
Reviewer: R.Manthey (Jena)

MSC:

34G10 Linear differential equations in abstract spaces
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References:

[1] L. V. Ovsyannikov, ”A singular operator in Banach scales,” Dokl. Akad. Nauk SSSR,163, No. 4, 819–822 (1965).
[2] F. Tréves, ”An abstract nonlinear Cauchy-Kovalevska theorem,” Trans. Amer. Math. Soc.,150, 77–92 (1970). · Zbl 0199.15803
[3] L. Nirenberg, Topics in Nonlinear Functional Analysis [Russian translation], Mir, Moscow (1977). · Zbl 0426.47034
[4] L. V. Ovsyannikov, ”A nonlinear Cauchy problem in the scale of Banach spaces,” Dokl. Akad. Nauk SSSR,200, No. 4, 769–792 (1971). · Zbl 0234.35018
[5] L. V. Ovsyannikov, Analytic Groups [in Russian], Novosibirsk. Univ., Novosibirsk (1972). · Zbl 0489.22021
[6] L. V. Ovsyannikov, N. I. Makarenko, V. I. Nalimov et al., Nonlinear Problems in the Theory of Surface and Internal Waves [in Russian], Nauka, Novosibirsk (1985).
[7] O. V. Kaptsov, ”Inversion of the canonical multiplication ofA-groups,” Dinamika Sploshn. Sredy (Novosibirsk),50, 74–84 (1981). · Zbl 0489.22020
[8] J. Sebastião e Silva, ”Su certe classi di spazi localmente convessi importanti per le applicazioni,” Matematika,1, No. 1, 60–77 (1957).
[9] H. Grauert and R. Remmert, Analytische Stellenalgebren [Russian translation], Nauka, Moscow (1988).
[10] S. S. Titov, ”Analyticity of one-parameter linear Lie-Bäcklund groups,” Differentsial’nye Uravneniya,36, No. 4, 699–702 (1990). · Zbl 0704.58057
[11] S. S. Titov, ”The Cauchy problem in special scales of Banach spaces,” in: Approximate Methods for Studying Nonlinear Problems of Continuum Mechanics [in Russian], IMM Ural’sk. Otdel. Ross. Akad. Nauk, Sverdlovsk, 1992, pp. 51–57. · Zbl 0827.35030
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