Global existence and stability results of a nonlinear swelling soil system with nonlinear source terms of variable exponent types

We consider a swelling porous-elastic system with two frictional damping and source terms. Those frictional damping and source terms are nonlinear and of variable exponents type. First, we prove that the system has a global solution by using the potential wells method. Then, we establish polynomial and exponential decay rates for the energy of the system by using the Multiplier method. All the existence and stability results are obtained under a natural condition on the parameters of the system and the variable exponents. We discuss our stability results based on the coefficients of the frictional damping terms and the strongest nonlinearity of the variable exponents. AMS Subject Classifiations: 93D20; 93D23; 35B40; 74F05.