Coupled second order evolution equations with memory effects and variable sign kernels

We consider a system of coupled second order integro-differential evolution equations in a Hilbert space, which is partially damped through memory effects. A global existence theorem regarding the solutions to its Cauchy problem is given, only under basic conditions that the memory kernels possess positive definite primitives but without nonnegative/decreasing assumptions. Following this, we find an approach to successfully obtain the stability of the system energy and various decay rates. Moreover, the abstract results are applied to several concrete systems in the real world, including the Timoshenko type. This is a joint work with Professor Ti-Jun Xiao (Fudan University) and Professor Jin Liang (Shanghai Jiaotong University)