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

  • C. Jendoubi Univ. Sfax, Tunisia


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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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.
Research articles