Regularity for evolution equations with nonlocal initial conditions

  • Pengyu Chen
  • Xuping Zhang
  • Yongxiang Li
Original Paper


This paper is concerned with the existence and uniqueness of global strong solutions for a class of semilinear evolution equations with nonlocal initial conditions on infinite interval. We assume that the linear part is a positive definite selfadjoint operator, and the nonlinear part satisfies some essential growth conditions. The discussions are based on analytic semigroups theory, the piecewise regular method and relevant fixed point theorem. The results obtained in this paper improve and extend some related conclusions on this topic. An example to illustrate the feasibility of our abstract results is also given.


Evolution equations Nonlocal initial condition Compact analytic semigroup Strong solution 

Mathematics Subject Classification

46B50 47H20 47J35 



The authors would like to thank the anonymous referees for their carefully reading the manuscript and very important comments and suggestions that improved the results and quality of this paper. P. Chen acknowledge support from NNSF of China (No. 11501455) and Key Project of Gansu Provincial National Science Foundation (No. 1606RJZA015). Y. Li acknowledge support from NNSF of China (No. 11661071).


  1. 1.
    Boucherif, A.: Semilinear evolution inclutions with nonlocal conditions. Appl. Math. Lett. 22(8), 1145–1149 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Byszewski, L.: Application of preperties of the right hand sides of evolution equations to an investigation of nonlocal evolution problems. Nonlinear Anal. 33(5), 413–426 (1998)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Byszewski, L.: Existence and uniqueness of a classical solutions to a functional-differential abstract nonlocal Cauchy problem. J. Math. Appl. Stoch. Anal. 12(1), 91–97 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Chen, P., Li, Y.: Monotone iterative technique for a class of semilinear evolution equations with nonlocal conditions. Results Math. 63(3), 731–744 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Corduneanu, C.: Principles of Differential and Integral Equations. Allyn and Bacon, Boston (1971)MATHGoogle Scholar
  6. 6.
    Fu, X., Ezzinbi, K.: Existence of solutions for neutral equations with nonlocal conditions. Nonlinear Anal. 54(2), 215–227 (2003)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840. Springer, New York (1981)CrossRefGoogle Scholar
  8. 8.
    Liang, J., Liu, J.H., Xiao, T.J.: Nonlocal Cauchy problems governed by compact operator families. Nonlinear Anal. 57(2), 183–189 (2004)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Liang, J., Liu, J., Xiao, T.J.: Nonlocal Cauchy problems for nonautonomous evolu tion equations. Commun. Pure. Appl. Anal. 5(3), 529–535 (2006)MathSciNetCrossRefGoogle Scholar
  10. 10.
    McKibben, M.: Discoving Evolution Equations with Applications, Vol. I Deterministic Models. Chapman and Hall/CRC Appl. Math. Nonlinear Sci. Ser. (2011)Google Scholar
  11. 11.
    Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, Berlin (1983)CrossRefMATHGoogle Scholar
  12. 12.
    Teman, R.: Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edn. Springer, New York (1997)CrossRefGoogle Scholar
  13. 13.
    Vrabie, I.I.: Existence in the large for nonlinear delay evolution inclutions with nonlocal initial conditions. J. Funct. Anal. 262(4), 1363–1391 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Xiang, X., Ahmed, N.U.: Existence of periodic solutions of semilinear evilution equations with time lags. Nonlinear Anal. 18(11), 1063–1070 (1992)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Xiao, T.J., Liang, J.: Existence of classical solutions to nonautonomous nonlocal parabolic problems. Nonlinear Anal. 63(5–7), 225–232 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Xue, X.: Nonlocal nonlinear differential equations with a measure of noncompactness in Banach spaces. Nonlinear Anal. 70(7), 2593–2601 (2009)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia 2017

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouPeople’s Republic of China

Personalised recommendations