Unstable manifolds of a class of delayed partial differential equations with nondense domain
Volume 118 / 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.
