# 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}$are densely defined closed linear operators on a Banach space. A Pazy-type characterization of the infinitely differentiable propagators of the Cauchy problem is obtained. Moreover, two related sufficient conditions are given. Citation: Ti-Jun Xiao, Jin Liang. Pazy-type characterization for differentiability of propagators of higher order Cauchy problems in Banach spaces. Discrete & Continuous Dynamical Systems - A, 1999, 5 (3) : 651-662. doi: 10.3934/dcds.1999.5.651  [1] Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181 [2] Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851 [3] T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors. Inverse Problems & Imaging, 2017, 11 (5) : 857-874. doi: 10.3934/ipi.2017040 [4] Jon Johnsen. Well-posed final value problems and Duhamel's formula for coercive Lax–Milgram operators. Electronic Research Archive, 2019, 27: 20-36. doi: 10.3934/era.2019008 [5] W. Layton, R. Lewandowski. On a well-posed turbulence model. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 111-128. doi: 10.3934/dcdsb.2006.6.111 [6] Shouming Zhou. The Cauchy problem for a generalized$b$-equation with higher-order nonlinearities in critical Besov spaces and weighted$L^p$spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967 [7] Huijun He, Zhaoyang Yin. On the Cauchy problem for a generalized two-component shallow water wave system with fractional higher-order inertia operators. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1509-1537. doi: 10.3934/dcds.2017062 [8] Yuanyuan Ren, Yongsheng Li, Wei Yan. Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2018, 17 (2) : 487-504. doi: 10.3934/cpaa.2018027 [9] Rinaldo M. Colombo, Mauro Garavello. A Well Posed Riemann Problem for the$p\$--System at a Junction. Networks & Heterogeneous Media, 2006, 1 (3) : 495-511. doi: 10.3934/nhm.2006.1.495 [10] 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 [11] 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 [12] 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 [13] Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete & Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127 [14] 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 [15] G. R. Cirmi, S. Leonardi. Higher differentiability for solutions of linear elliptic systems with measure data. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 89-104. doi: 10.3934/dcds.2010.26.89 [16] Mikil Foss, Joe Geisbauer. Higher differentiability in the context of Besov spaces for a class of nonlocal functionals. Evolution Equations & Control Theory, 2013, 2 (2) : 301-318. doi: 10.3934/eect.2013.2.301 [17] Hongqiu Chen. Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 397-429. doi: 10.3934/dcds.2018019 [18] Mohammed AL Horani, Mauro Fabrizio, Angelo Favini, Hiroki Tanabe. Fractional Cauchy problems and applications. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2259-2270. doi: 10.3934/dcdss.2020187 [19] Micol Amar, Andrea Braides. A characterization of variational convergence for segmentation problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (3) : 347-369. doi: 10.3934/dcds.1995.1.347 [20] Stefan Kindermann. Convergence of the gradient method for ill-posed problems. Inverse Problems & Imaging, 2017, 11 (4) : 703-720. doi: 10.3934/ipi.2017033

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