Zelik, S. V. The attractor of a quasilinear hyperbolic equation with dissipation in \(\mathbb{R}^n\): Dimension and \(\varepsilon\)-entropy. (English. Russian original) Zbl 0974.35016 Math. Notes 67, No. 2, 248-251 (2000); translation from Mat. Zametki 67, No. 2, 304-308 (2000). 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. Reviewer: Victor Sharapov (Volgograd) Cited in 1 ReviewCited in 2 Documents MSC: 35B41 Attractors 35L70 Second-order nonlinear hyperbolic equations 37L30 Attractors and their dimensions, Lyapunov exponents for infinite-dimensional dissipative dynamical systems 35L15 Initial value problems for second-order hyperbolic equations Keywords:Kolmogorov \(\varepsilon\)-entropy; fractal dimension PDFBibTeX XMLCite \textit{S. V. Zelik}, Math. Notes 67, No. 2, 248--251 (2000; Zbl 0974.35016); translation from Mat. Zametki 67, No. 2, 304--308 (2000) Full Text: DOI References: [1] Babin, A. V.; Vishik, M. I., Attractors of Evolution Equations (1989), Moscow: Nauka, Moscow · Zbl 0804.58003 [2] Ghidaglia, J.-M.; Temam, R., J. Math. Pures Appl., 66, 273-319 (1987) · Zbl 0572.35071 [3] J. K. Hale,Asymptotic Behaviour of Dissipative Systems, Math. Surveys Monographs, Vol. 25, Amer. Math. Soc., Providence, R.I. (1987). · Zbl 0657.35014 [4] Feireisl, E., Differential Integral Equations, 9, 1147-1156 (1996) · Zbl 0858.35084 [5] Ginibre, J.; Soffer, A.; Velo, G., J. Funct. Anal., 110, 96-130 (1992) · Zbl 0813.35054 [6] Kolmogorov, A. N.; Tikhomirov, V. M., Uspekhi Mat. Nauk [Russian Math. Surveys], 14, 2-86, 3-86 (1959) · Zbl 0090.33503 [7] Vishik, M. I.; Chepyzhov, V. V., Mat. Sb. [Russian Acad. Sci. Sb. Math.], 189, 2, 81-110 (1998) · Zbl 0915.35056 [8] Collet, P.; Eckmann, J., Commun. Math. Phys., 200, 699-722 (1999) · Zbl 0920.35071 [9] Zelik, S. V., Mat. Zametki [Math. Notes], 65, 6, 941-943 (1999) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.