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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.

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
Full Text: DOI
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