$L^{p}-L^{q}$ decay estimates for the Cauchy problem of linear thermoelastic systems with second sound in one space variable

$L^p-L^q$ decay estimates of solutions to the Cauchy problem of linear thermoelastic systems with second sound in one space variable will be studied in this paper. First, by dividing the frequency of phase space of the Fourier transformation into different regions, the asymptotic behavior of characteristic roots of the coefficient matrix is obtained by carefully analyzing the effect of the different regions. Second, with the help of the information on the characteristic roots and by using the interpolation theorem, the $L^{p}-L^{q}$ decay estimate of solutions to the Cauchy problem of the linear thermoelastic system with second sound in one space variable is obtained.