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Attractors for the generalized Benjamin-Bona-Mahony equation. (English) Zbl 0934.35151

The authors deal with the periodic initial-boundary value problem \[ u_t-a\Delta u_t-b\Delta u+\nabla\cdot F(u)=h(x),\quad u(x+L_ie_i,t) =u(x,t),\;u(x,0)=u_0(x), \] where \(x\in\mathbb{R}^n\); \(t\geq 0\); \(L_i,a,b>0\); \(e_i\) is the canonical basis; \(\nabla\cdot F=\Sigma(\partial/\partial x_i)F_i\); \(F_i(0) =0\); \(|dF_i(s)/ds |\leq C(1+|s|^m)\); \(h\in\dot L_2\) (i.e., \(h\in L_2\) and \(\int h dx=0\) over the periodicity domain); \(u_0\in\dot H^1\) (i.e., moreover \(\int u_0dx=0)\).
There exists a unique solution \(u\in C(\mathbb{R}^+,\dot H^1)\) which generates a semigroup \(V_t:\dot H^1\to\dot H^1(t\geq 0)\). This semigroup has a global attractor \({\mathcal M}\) (a minimal closed set \(M\subset\dot H^1\) which attracts each bounded subset of \(\dot H^1)\), which is compact, invariant, and connected in \(\dot H^1\). Moreover \({\mathcal M}\) is bounded in \(H^2\cap\dot H^1\) and of a finite fractal dimension.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems
35B10 Periodic solutions to PDEs
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