Estimate Of Entropy For The Attractor Of Strongly Damped Wave Equation

In this paper, we mainly consider the estimate of entropy of the attractor for infinitedimension dynamical system in unbounded domain.The basic problem of infinite dimension dynamical system is to consider the existenceof the global attractor of the solution semigroup for dissipative partial differential equationin appropriate Banach space or complete metric space and reveal the geometric topologicalproperties of the global attractor. After proving the existence of the global attractor,a naturalquestion is to study its features for further analysis and consequently we expect to be able tolearn more about the essential characteristics of the infinite dimensional dynamical system.In the nearly two decades, characterization of the global attractor of dissipative system foranalytic properties and geometric topology properties has always been the people’s atten-tion. For example, V.Chepyzhov、M.A.Efendiev、A.Ilyin、M.Vishik and others havedone much work about estimate of dimension and complexity of the attractor.When the area of the phase space is bounded domain, the attractors of a variety ofequations such as reaction-diffusion equation have finite Hausdorff and fractal dimension,in contrast to the bounded domains, the case of the unbounded domains is more complex. Inthis case, we can not get the ideal conclusion that dynamical system limited to the attractorhas finite dimension. In other words, fractal dimension is not a convenient description ofthe quantitative characteristics of the attractor size. However, the attractor is obviously“thinner”than the phase space. In some sense, we can use fewer degrees of freedomto limit system. So people use Kolmogorov ε-entropy to describe the “thickness”of theattractor.We take strongly damped semilinear wave equation for example,i.e.(?)Because the area of the phase space is unbounded domain,we use locally uniform space asthe phase space for including constant solutions,traveling wave solutions and other kinds ofspecial forms of solutions and estimating the upper and lower bounds of the entropy of theattractors.