## 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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### References:

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