We consider the following semilinear dissipative wave equations with linear memory on the unbounded domain RN utt + ±ut ? k(0)á(x)￠u ?Z 10 k0(s)á(x)￠u(t ? s)ds + ?f(u) = h(x); (x; t) 2 RN ￡ R+; u(x; t) = u0(x; t); x 2 RN; t · 0: Where N ? 3, ± > 0, k(0), k(1) > 0, k0(s) · 0, 8s 2 R+ and u0(x; t) is the initial data. The energy space X0 = D1;2(RN)￡L2 g (RN)￡L2 1(R+;D1;2(RN)) is introduced, to overcome the di±culties related with the non-compactness of operators, which arise in unbounded domains. The estimates on the Hausdor? and fractal dimension are in terms of given parameters, due to an asymptotic estimate for the eigenvalues a of the eigenvalue problem ?á(x)￠u = au; x 2 RN.