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.

 

 

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Published
17.06.2020
How to Cite
JendoubiC. “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