# American Institute of Mathematical Sciences

July  1999, 5(3): 651-662. doi: 10.3934/dcds.1999.5.651

## Pazy-type characterization for differentiability of propagators of higher order Cauchy problems in Banach spaces

 1 Department of Mathematics, University of Science and Technology of China, Hefei 230026, China, China

Received  March 1998 Revised  May 1999 Published  May 1999

Of concern is the differentiability of the propagators of higher order Cauchy problem

$u^{(n)}(t) +\sum_{k=0}^{n-1}A_ku^{(k)}(t)=0, t\geq 0,$

$u^{(k)}(0) = u_k, 0\leq k\leq n-1, where$A_0, A_1,\ldots, A_{n-1}$are densely defined closed linear operators on a Banach space. A Pazy-type characterization of the infinitely differentiable propagators of the Cauchy problem is obtained. Moreover, two related sufficient conditions are given. Citation: Ti-Jun Xiao, Jin Liang. Pazy-type characterization for differentiability of propagators of higher order Cauchy problems in Banach spaces. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 651-662. doi: 10.3934/dcds.1999.5.651  [1] Doria Affane, Mustapha Fateh Yarou. Well-posed control problems related to second-order differential inclusions. Evolution Equations and Control Theory, 2022, 11 (4) : 1229-1249. doi: 10.3934/eect.2021042 [2] Belkacem Said-Houari. Global well-posedness of the Cauchy problem for the Jordan–Moore–Gibson–Thompson equation with arbitrarily large higher-order Sobolev norms. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4615-4635. doi: 10.3934/dcds.2022066 [3] Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181 [4] Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851 [5] T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors. Inverse Problems and Imaging, 2017, 11 (5) : 857-874. doi: 10.3934/ipi.2017040 [6] Jon Johnsen. Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators. Electronic Research Archive, 2019, 27: 20-36. doi: 10.3934/era.2019008 [7] W. Layton, R. Lewandowski. On a well-posed turbulence model. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 111-128. doi: 10.3934/dcdsb.2006.6.111 [8] Shouming Zhou. The Cauchy problem for a generalized$b$-equation with higher-order nonlinearities in critical Besov spaces and weighted$L^p$spaces. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967 [9] Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062 [10] Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1221-1232. doi: 10.3934/dcdss.2021093 [11] Rinaldo M. Colombo, Mauro Garavello. A Well Posed Riemann Problem for the$p\$--System at a Junction. Networks and Heterogeneous Media, 2006, 1 (3) : 495-511. doi: 10.3934/nhm.2006.1.495 [12] Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990 [13] John Baxley, Mary E. Cunningham, M. Kathryn McKinnon. Higher order boundary value problems with multiple solutions: examples and techniques. Conference Publications, 2005, 2005 (Special) : 84-90. doi: 10.3934/proc.2005.2005.84 [14] Carlos Durán, Diego Otero. The projective symplectic geometry of higher order variational problems: Minimality conditions. Journal of Geometric Mechanics, 2016, 8 (3) : 305-322. doi: 10.3934/jgm.2016009 [15] Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127 [16] Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451 [17] Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 [18] G. R. Cirmi, S. Leonardi. Higher differentiability for solutions of linear elliptic systems with measure data. Discrete and Continuous Dynamical Systems, 2010, 26 (1) : 89-104. doi: 10.3934/dcds.2010.26.89 [19] Mikil Foss, Joe Geisbauer. Higher differentiability in the context of Besov spaces for a class of nonlocal functionals. Evolution Equations and Control Theory, 2013, 2 (2) : 301-318. doi: 10.3934/eect.2013.2.301 [20] Hongqiu Chen. Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 397-429. doi: 10.3934/dcds.2018019

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