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Inertial manifolds for retarded second order in time evolution equations. (English) Zbl 1023.35087
The author investigates a nonlinear damped wave equation with retardation: $\partial^2_t u+2\varepsilon \partial_tu+Au =B(u_t),\quad \varepsilon> 0\tag{1}$ with initial data $$u(\theta)= u^0(\theta)$$, $$\theta\in [-r,0]$$, $$(\partial_tu) (0)=u^1$$. Here $$\varepsilon >0$$ is a not necessarily small parameter and $$A$$ a selfadjoint positive operator on a Hilbert space $$H$$ with spectrum $0<\mu_0 \leq\mu_1\leq \dots,\lim \mu_k=\infty,\;k\to\infty.$ The operator $$A$$ gives rise to fractional pover spaces $$D(A^\alpha)$$, $$\alpha\in [0, \tfrac 12]$$ endowed with norm $$\|\;|_\alpha= \|A^\alpha\|$$ and to the function space $$C_\alpha=C( [-r,0];D(A^\alpha))$$; for $$u\in C([-r,\infty]$$; $$D(A^\alpha))$$ and $$t\geq 0$$, $$u_t\in C_\alpha$$ is given by $$(u_t)(\theta)= u(t-\theta)$$, $$\theta\in [-r,0]$$. On $$C_\alpha$$ one has a norm according to $|v|_\alpha= \sup\bigl\|A^\alpha v(\theta) \bigr\|,\;\theta\in [-r,0].$ The nonlinearity $$B(\cdot)$$ in (1) has the form: $B(v)=B_0 \bigl(v(0)\bigr)+ B_1(v),\;v\in C_\alpha,$ where $$B_0$$ maps $$D(A^\alpha)$$ into $$H$$ while $$B_1$$ maps $$C_\alpha$$ into $$H$$. The retardation is thus built into $$B_1$$. One assumes Lipschitz conditions: \begin{aligned} \bigl\|B_0(w_1)- B_0(w_2)\bigr\|\leq M_0 \|w_1-w_2\|_\alpha,\quad & w_j\in D(A^\alpha),\\ \bigl\|B_1(v_1)-B_1 (v_2)\bigr\|\leq M_1|v_1-v_2|_\alpha,\quad & v_j\in C_\alpha. \end{aligned} Equation (1) is then transformed into a system $\partial_t U+A'U= B'(U_t),\;U_0=(u^0,u^1)\tag{2}$ where $$A'(u,v)=(-v,Au+2\varepsilon v)$$, $$B'(u,v) =(0,B(u_t))$$. Eigenvalues and eigenvectors of $$A'$$ are now given by $\lambda^\pm_n=\varepsilon \pm(\varepsilon^2-\mu_n)^{\tfrac 12},\;f_n^\pm= (e_n,-\lambda^\pm_n e_n),\;Ae_n=\mu_ne_n.$ For some $$N$$, specified below, one assumes $$\varepsilon^2> \mu_{N+1}$$. The author now asserts the existence of an inertial manifold for (2) provided that a somewhat involved spectral gap condition is satisfied. It is assumed that $$N$$ above is such that besides $$\varepsilon^2<\mu_{N+1}$$, the following holds: $\lambda^-_{N+1}-\lambda_N^-<c_0 \cdot\mu_{N+1}^{-\tfrac 12}\max\bigl(1,\mu_{N+1}^{\tfrac 12}(\varepsilon^2-\mu_{ N+1})^{-\tfrac 12}\bigr)(M_0+M_1)$ where $$c_0=8(5-17^{\tfrac 12})^{-\tfrac 12}$$. Under these assumptions and provided that the gap parameter $$r$$ is sufficiently small, Theroem 3.1 of the paper asserts the existence of an inertial manifold for (2). No examples are given.

##### MSC:
 35R10 Functional partial differential equations 37D10 Invariant manifold theory for dynamical systems 37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems 35B42 Inertial manifolds 35L70 Second-order nonlinear hyperbolic equations
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