在R^N上的具有线性记忆项的非线性反应扩散方程的吸引子的Hausdorff维数和分形维数估计

Estimates on the Hausdorff and fractal dimension of attractors fornonlinear reaction diffusion equations with linear memory in R^N

在无界区域RN上考虑了一类在Coleman-Gwrtin理论中经常出现的具有线性记忆项(用卷积项来表示,反映一个或多个变量的过去历史变化情况)的非线性热传导积分-微分方程ut-Δu-∫0∞k(s)Δu(t-s)ds=f(x,u).对非线性项f(x;u)施加负指数型的条件,把方程改述成历史空间框架下,对相关解的半群的整体吸引子估计了Hausdoff.维数和分形维数的上界.

We consider a class of nonlinear heat conduction integro-differential equations with linear memory terms,expressed by convolution integrals,which account for the past history of a variableut-Δu-∫∞0k（s）Δu（t-s）ds=f（x,u）,x∈RN;u（0）=u0.arising in the Coleman-Gurtin＇s theory.Under the negative exponential type condition for the nonlinear term f（x;u）,we rephras the equation within the history space framework and estimate the Hausdorff.and fractal dimension of the attractor for the related solution semigroup.