A+ CATEGORY SCIENTIFIC UNIT

PDF files of articles are only available for institutions which have paid for the online version upon signing an Institutional User License.

Unstable manifolds of a class of delayed partial differential equations with nondense domain

Volume 118 / 2016

Chiraz Jendoubi Annales Polonici Mathematici 118 (2016), 181-208 MSC: 34D09, 34G20, 35B40, 47D06. DOI: 10.4064/ap3913-11-2016 Published online: 23 December 2016

Abstract

We present an unstable manifold theory for the abstract delayed semilinear Cauchy problem with nondense domain $$ \frac {du}{dt}=(A+B(t))u(t)+f(t,u_t),\hskip 1em t\in \mathbb {R}, $$ 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 $\phi , \psi \in \mathcal {C}:=C([-r,0],X)$.

To reach our goal, we rely mainly on extrapolation theory. First, we develop a new variation of constants formula adapted to our problem. Then, using the characterization of exponential dichotomy, the properties of admissible spaces, the Lyapunov–Perron method as well as useful technical structures we prove the existence of an unstable manifold for our solutions. We also state an exponential attractiveness result concerning the unstable manifold. For illustration, we give an example.

Authors

  • Chiraz JendoubiFaculty of Sciences
    University of Sfax
    Sfax, Tunisia
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image