The attractor of a quasilinear hyperbolic equation with dissipation in \(\mathbb{R}^n\): Dimension and \(\varepsilon\)-entropy
In the article a hyperbolic equation \[ \partial^2_tu+ \gamma\partial_tu- \Delta_xu+ f(u)= g(x),\quad \gamma> 0,\quad u|_{t=0}= u_0;\quad\partial u|_{t=0}= u_0'\tag{1} \] is studied. The function \(f\) is assumed to be \(f(u)= \lambda_0 u+ f_1(u)+ f_2(u)\), \(\lambda_0>0\), where \(f_1\) has some restrictions on the growth rate with respect to \(u\) and \(f_2\) as well as its derivative are bounded. For the phase space of a problem (1) the space \(E_b= W^{1,2}_b(\mathbb{R}^n)\times L^2_b(\mathbb{R}^n)\), \(\xi_n\equiv (u,\partial_t u)\in E_b\) is chosen. Here \(W^{l,p}_b(\mathbb{R}^n)= \{n\in{\mathcal D}'(\mathbb{R}^n):\|u\|_{W^p_l(B^1_{x_0})}< \infty\}\), where \(W^p_l\) is the Sobolev space and \(B^R_{x_0}\) is the ball of radius \(R\) centered at \(x_0\). If \(g{\i}L^2_b(\mathbb{R}^n)\), then for all \(\xi_u(0)= (u_0, u_0')\in E_b\) there exists a unique solution \(\xi_u(t)\in E_b\) for \(t\geq 0\). Thus the problem (1) generates the semigroup \(S_t: E_b\to E_b\), \(S_t\xi_u(0)= \xi_u(t)\). An attractor \({\mathcal A}\subset E_b\) of the semigroup \(S_t\) is defined. It is shown that the semigroup \(S_t\) has an attractor \({\mathcal A}\), \({\mathcal A}\) is closed but not compact in \(E_b\), and the Kolmogorov \(\varepsilon\)-entropy of \({\mathcal A}\) admits some estimates. Hence the restriction of the attractor to any ball \(B^R_{x_0}\) is of finite fractal (entropy) dimension. If \(f_2\equiv 0\), then the attractor \({\mathcal A}\) is a compact set in \(E_b\) and its fractal dimension is finite. The results are given without proofs.
scientific article; zbMATH DE number 1517710