On the theory of integral manifolds for some delayed partial differential equations with nondense domain

  • C. Jendoubi Univ. Sfax, Tunisia

Abstract

UDC 517.9

Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation

$$\frac{du}{dt}=(A+B(t))u(t)+f(t,u_t),\quad t\in\mathbb{R},\tag{1}$$

where $(A,D(A))$ satisfies the Hille–Yosida condition, $(B(t))_{t\in\mathbb{R}}$ is a family of operators in $\mathcal{L}(\overline{D(A)},X)$ satisfying some measurability and boundedness conditions, and the nonlinear forcing term $f$ satisfies $\|f(t,\phi)-f(t,\psi)\|\leq \varphi(t)\|\phi-\psi\|_{\mathcal{C}}$;  here, $\varphi$ belongs to some admissible spaces and $\phi,$ $\psi\in\mathcal{C}:=C([-r,0],X)$. We first present an exponential convergence result between the stable manifold and every mild solution of (1).  Then we prove the existence of center-unstable manifolds for such solutions.

Our main methods are invoked by the extrapolation theory and the Lyapunov–Perron method based on the admissible functions properties.

 

 

References

N. N. Bogoliubov, Yu. A. Mitropolsky, The method of integral manifolds in nonlinear mechanics, Contrib. Different. Equat., 2, 123 – 196 (1963).

L. Boutet de Monvel, I. D. Chueshov, A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Anal., 34, 907 – 925 (1998) https://doi.org/10.1016/S0362-546X(97)00569-5 DOI: https://doi.org/10.1016/S0362-546X(97)00569-5

S. N. Chow, K. Lu, Invariant manifolds for ows in Banach spaces, J. Different. Equat., 74, 285 – 317 (1988) https://doi.org/10.1016/0022-0396(88)90007-1 DOI: https://doi.org/10.1016/0022-0396(88)90007-1

P. Constantin, C. Foias, B. Nicolaenko, R. Temam,Integral manifolds and inertial manifolds for dissipative partial differential equations, Springer-Verlag, New York (1989) https://doi.org/10.1007/978-1-4612-3506-4 DOI: https://doi.org/10.1007/978-1-4612-3506-4

G. Da Prato, E. Sinestrari, Differential operators with non-dense domains, Ann. Scuola Norm. Super. Pisa Cl. Sci., 14, 285 – 344 (1987) http://www.numdam.org/item?id=ASNSP_1987_4_14_2_285_0

T. V. Duoc, N. T. Huy, Integral manifolds and their attraction property for evolution equations in admissible function spaces, Taiwanese J. Math., 16, 963 – 985 (2012) https://doi.org/10.11650/twjm/1500406669 DOI: https://doi.org/10.11650/twjm/1500406669

T. V. Duoc, N. T. Huy, Integral manifolds for partial functional differential equations in admissible spaces on a half line, J. Math. Anal. and Appl., 411, 816 – 828 (2014) https://doi.org/10.3934/dcdsb.2015.20.2993 DOI: https://doi.org/10.3934/dcdsb.2015.20.2993

T. V. Duoc, N. T. Huy, Unstable manifolds for partial functional differential equations in admissible spaces on the whole line, Vietnam J. Math., 32, 37 – 55 (2015) https://doi.org/10.1007/s10013-016-0234-7 DOI: https://doi.org/10.1007/s10013-016-0234-7

G. Guhring, F. Rabiger, Asymptotic properties of mild solutions for nonautonomous evolution equations with appli- cations to retarded differential equations, J. Abstr. and Appl. Anal., 4, No 3, 169 – 194 (1999) https://doi.org/10.1155/S1085337599000214 DOI: https://doi.org/10.1155/S1085337599000214

M. W. Hirsch, C. C. Pugh, M. Shub, Invariant manifolds, Springer-Verlag, Berlin; Heidelberg (1977). DOI: https://doi.org/10.1007/BFb0092042

N. T. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235, 330 – 354 (2006) https://doi.org/10.1016/j.jfa.2005.11.002

N. T. Huy, Stable manifolds for semi-linear evolution equations and admissibility of function spaces on a half-line, J. Math. Anal. and Appl., 354, 372 – 386 (2009) https://doi.org/10.1016/j.jfa.2005.11.002 DOI: https://doi.org/10.1016/j.jfa.2005.11.002

N. T. Huy, Invariant manifolds of admissible classes for semi-linear evolution equations, J. Differen. Equat., 246, 1822 – 1844 (2009) https://doi.org/10.1016/j.jde.2008.10.010 DOI: https://doi.org/10.1016/j.jde.2008.10.010

C. Jendoubi, Unstable manifolds of a class of delayed partial differential equations with nondense domain, Ann. Polon. Math., 181 – 208 (2016) https://doi.org/10.4064/ap3913-11-2016 DOI: https://doi.org/10.4064/ap3913-11-2016

C. Jendoubi, Integral manifolds of a class of delayed partial differential equations with nondense domain, Numer. Funct. Anal. and Optim., 38, 1024 – 1044 (2017) https://doi.org/10.1080/01630563.2017.1309665 DOI: https://doi.org/10.1080/01630563.2017.1309665

Z. Liu, P. Magal, S. Ruan, Center-unstable manifolds for non-densely de ned Cauchy problems and applications to stability of Hopf bifurcation, Canad. Appl. Math. Quart., 20, 135 – 178 (2012) https://www.math.u-bordeaux.fr/~pmagal100p/papers/LMR-CAMQ-13.pdf

L. Maniar, Stability of asymptotic properties of Hille – Yosida operators under perturbations and retarded differential equations, Quaest. Math., 28, 39 – 53 (2005)https://doi.org/10.2989/16073600509486114 DOI: https://doi.org/10.2989/16073600509486114

N. V. Minh, J. Wu, Invariant manifolds of partial functional differential equations, J. Different. Equat., 198, 381 – 421 (2004) https://doi.org/10.1016/j.jde.2003.10.006 DOI: https://doi.org/10.1016/j.jde.2003.10.006

J. D. Murray, Mathematical biology I: an introduction, Springer-Verlag, Berlin (2002) ISBN 978-3-662-08542-4

J. D. Murray, Mathematical biology II: spatial models and biomedical applications, Springer-Verlag, Berlin (2003) xxvi+811 pp. ISBN: 0-387-95228-4

A. Rhandi, Extrapolation methods to solve non-autonomous retarded partial differential equations, Stud. Math., 126, No 3, 219 – 233 (1998) https://doi.org/10.4064/sm-126-3-219-233 DOI: https://doi.org/10.4064/sm-126-3-219-233

H. R. Thieme, Semi ows generated by Lipschitz perturbations of nondensely de ned operators, Different. Integral Equat., 3, 1035 – 1066 (1990).

H. R. Thieme, ”Integrated semigroups” and integrated solutions to abstract Cauchy problems, J. Math. Anal. and Appl., 152, 416 – 447 (1990)https://doi.org/10.1016/0022-247X(90)90074-P DOI: https://doi.org/10.1016/0022-247X(90)90074-P

Published
17.06.2020
How to Cite
Jendoubi, C. “On the Theory of Integral Manifolds for Some Delayed Partial Differential Equations With Nondense Domain”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 6, June 2020, pp. 776-89, doi:10.37863/umzh.v72i6.6020.
Section
Research articles