American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Explicit quasiconvexification for some cost functionals depending on derivatives of the state in optimal designing
  • DCDS Home
  • This Issue
  • Next Article
    Estimates on the dimension of a global attractor for a semilinear dissipative wave equation on $\mathbb R^N$
November  2002, 8(4): 953-966. doi: 10.3934/dcds.2002.8.953

PFDE with nonautonomous past

Simon Brendle 1, and  Rainer Nagel 2,

1. 

Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany
2. 

University of Tübingen, Mathematics Institute, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Received  April 2001 Revised  February 2002 Published  July 2002

We study Cauchy problems associated to partial differential equations with infinite delay where the history function is modified by an evolution family. Using sophisticated tools from semigroup theory such as evolution semigroups, extrapolation spaces, or the critical spectrum, we prove well-posedness and characterize the asymptotic behavior of the solution semigroup by an operator-valued characteristic equation.
Keywords: nonautonomous Cauchy problem, evolution semigroups, stability., pde with delay.
Mathematics Subject Classification: 34K06, 35R10, 47D0.
Citation: Simon Brendle, Rainer Nagel. PFDE with nonautonomous past. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 953-966. doi: 10.3934/dcds.2002.8.953
[1]

Jin Liang, James H. Liu, Ti-Jun Xiao. Nonlocal Cauchy problems for nonautonomous evolution equations. Communications on Pure and Applied Analysis, 2006, 5 (3) : 529-535. doi: 10.3934/cpaa.2006.5.529
[2]

George Avalos. Strong stability of PDE semigroups via a generator resolvent criterion. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 207-218. doi: 10.3934/dcdss.2008.1.207
[3]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems and Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791
[4]

Tomás Caraballo, Leonid Shaikhet. Stability of delay evolution equations with stochastic perturbations. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2095-2113. doi: 10.3934/cpaa.2014.13.2095
[5]

Ismael Maroto, Carmen Núñez, Rafael Obaya. Exponential stability for nonautonomous functional differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3167-3197. doi: 10.3934/dcdsb.2017169
[6]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171
[7]

Qiang Li, Mei Wei. Existence and asymptotic stability of periodic solutions for neutral evolution equations with delay. Evolution Equations and Control Theory, 2020, 9 (3) : 753-772. doi: 10.3934/eect.2020032
[8]

Cunming Liu, Jianli Liu. Stability of traveling wave solutions to Cauchy problem of diagnolizable quasilinear hyperbolic systems. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4735-4749. doi: 10.3934/dcds.2014.34.4735
[9]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469
[10]

Yuri Latushkin, Valerian Yurov. Stability estimates for semigroups on Banach spaces. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5203-5216. doi: 10.3934/dcds.2013.33.5203
[11]

Genni Fragnelli, Dimitri Mugnai. Nonlinear delay equations with nonautonomous past. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1159-1183. doi: 10.3934/dcds.2008.21.1159
[12]

Everaldo de Mello Bonotto, Matheus Cheque Bortolan, Rodolfo Collegari, José Manuel Uzal. Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (9) : 4645-4661. doi: 10.3934/dcdsb.2020306
[13]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045
[14]

Christian Pötzsche, Evamaria Russ. Topological decoupling and linearization of nonautonomous evolution equations. Discrete and Continuous Dynamical Systems - S, 2016, 9 (4) : 1235-1268. doi: 10.3934/dcdss.2016050
[15]

V. Varlamov, Yue Liu. Cauchy problem for the Ostrovsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (3) : 731-753. doi: 10.3934/dcds.2004.10.731
[16]

Mauro Garavello, Paola Goatin. The Cauchy problem at a node with buffer. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 1915-1938. doi: 10.3934/dcds.2012.32.1915
[17]

Adrien Dekkers, Anna Rozanova-Pierrat. Cauchy problem for the Kuznetsov equation. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 277-307. doi: 10.3934/dcds.2019012
[18]

Haifeng Hu, Kaijun Zhang. Stability of the stationary solution of the cauchy problem to a semiconductor full hydrodynamic model with recombination-generation rate. Kinetic and Related Models, 2015, 8 (1) : 117-151. doi: 10.3934/krm.2015.8.117
[19]

Angkana Rüland, Eva Sincich. Lipschitz stability for the finite dimensional fractional Calderón problem with finite Cauchy data. Inverse Problems and Imaging, 2019, 13 (5) : 1023-1044. doi: 10.3934/ipi.2019046
[20]

Arne Ogrowsky, Björn Schmalfuss. Unstable invariant manifolds for a nonautonomous differential equation with nonautonomous unbounded delay. Discrete and Continuous Dynamical Systems - B, 2013, 18 (6) : 1663-1681. doi: 10.3934/dcdsb.2013.18.1663

2021 Impact Factor: 1.588

Tools

Metrics

  • PDF downloads (345)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy