ON THE DECAY RATES FOR A ONE-DIMENSIONAL POROUS ELASTICITY SYSTEM WITH PAST HISTORY

This paper studies a porous elasticity system with past history {rho u(tt) - mu u(xx) - b phi(x) = 0, J phi(tt) - delta phi(xx) + bu(x) + xi phi + integral(infinity)(0) g(s)phi(xx)(t-s)ds = 0. By introducing a new variable, we establish an explicit and a general decay of energy for the case of equal-speed wave propagation as well as for the nonequal-speed case. To establish our results, we mainly adopt the method developed by Guesmia, Messaoudi and Soufyane [Electron. J. Differ. Equa. 2012(2012), 1-45] and some properties of convex functions developed by Alabau-Boussouira and Cannarsa [C. R. Acad. Sci. Paris Ser. I, 347(2009), 867-872], Lasiecka and Tataru [Differ. Inte. Equa., 6(1993), 507-533]. In addition we remove the assumption that b is positive constant in [J. Math. Anal. Appl., 469(2019), 457-471] and hence improve the result.