We prove the existence of integral (stable, unstable, center) manifolds of admissible classes for the solutions to the semilinear integral equation $u(t) = U(t,s)u(s) + ∫_s^t U(t,ξ)f(ξ,u(ξ))dξ$ when the evolution family $(U(t,s))_{t≥s}$ has an exponential trichotomy on a half-line or on the whole line, and the nonlinear forcing term f satisfies the (local or global) φ-Lipschitz conditions, i.e., ||f(t,x)-f(t,y)|| ≤ φ(t)||x-y|| where φ(t) belongs to some classes of admissible function spaces. These manifolds are formed by trajectories of the solutions belonging to admissible function spaces which contain wide classes of function spaces like function spaces of $L_p$ type, the Lorentz spaces $L_{p,q}$ and many other function spaces occurring in interpolation theory. Our main methods involve the Lyapunov-Perron method, rescaling procedures, and techniques using the admissibility of function spaces.
2
Dostęp do pełnego tekstu na zewnętrznej witrynie WWW
We prove the existence and conditional stability of periodic solutions to semilinear evolution equations of the form u̇ = A(t)u + g(t,u(t)), where the operator-valued function t ↦ A(t) is 1-periodic, and the operator g(t,x) is 1-periodic with respect to t for each fixed x and satisfies the φ-Lipschitz condition ||g(t,x₁) - g(t,x₂)|| ≤ φ(t)||x₁-x₂|| for φ(t) being a real and positive function which belongs to an admissible function space. We then apply the results to study the existence, uniqueness and conditional stability of periodic solutions to the above semilinear equation in the case that the family $(A(t))_{t≥0}$ generates an evolution family having an exponential dichotomy. We also prove the existence of a local stable manifold near the periodic solution in that case.
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.