Existência, unicidade e estabilidade de solução para um problema termoelástico hiperbólico com domínio não limitado em R

Abstract

Consider the Cauchy Problem which describes the dynamics of linear rafters vibrations in R subjected to thermal e ects modeled by the Cattaneo law. We will focus our attention on obtaining the existence and uniqueness of the solution and analyzing the asymptotic behavior of such a solution. In the rst part we will prove the existence and uniqueness of solutions for the thermoelastic model. 8> ><>>: utt + uxxxx ����� uxxtt + xx = 0; em R [0;+1) t + kqx ����� uxxt = 0; em R [0;+1) qt + q + k x = 0; em R [0;+1) : With initial conditions u(x; 0) = u0(x); ut(x; 0) = u1(x); (x; 0) = 0(x); q(x; 0) = q0(x): In the second part we nd a decay rate for total energy E(t) = 1 2 ZR u2 t + u2 tx + u2 xx + 2 + q2 dx associated with the model described above.