Estimates On The Dimension Of Attractors For Nonlinear Heat Conduction Equations With Linear Memory In R~N

We consider a class of nonlinear heat conduction integro-differential equations with linear memory terms, expressed by convolution integrals, wich account for the past history of one or more variables, arising in the Coleman-Gurtin’s theoryThese equations are models heat propagation in homogeneous isotropic heat conductor with hereditary memory. Here, the classical Fourier law ruling the heat flux is replaces by the more physical constitutive relation devised in the seminal paper of B.D.Coleman and Gurtin[l], based in the key assumption that the heat flux evolution if influenced by the past history of the temperature gradient. In that case, u represents of the temperature variation field relative to the equilibrium reference value,/is a time independent external heat supply with some assumptions:d2fδ2f/δs/(x,s)≤C0(M)for(?)|x|≤M>0,x∈RN with C0(t):Râ†'R+.And there exists C(x),D(x):Rnâ†'R. such that f(x,s)s≤C(x)|s|2+D(x)||s|,for (?)s∈R,xg RN, and the memory kernelμ{s)=-k’(s),has to comply with the exponential decay assumptionμ(s)+δμ(s)≤0for (?)s∈R+.In order to deal with memory term we introduce the memory spaces M0, and the new variable η=η’(s):[0,∞]×R+â†'R with ηt(x,s)=(?)s0u(x,t-y)dy.We prove the continuous dependence of solutions with respect of initial data and making use of the result of the continuous dependence get the semigroup T(t) is uniformly differentiable. And using the operatorâ–³+m(x)/has a negative exponential type, and Liouville formula, we can estimate the Hausdorff and fractal dimension of the attractor for the related solution semigroup.