On the existence and stability of a nonlinear wave system with variable exponents

Problems with variable exponents have attracted a great deal of attention lately and various existence, nonexistence and stability results have been established. The importance of such problems has manifested due to the recent advancement of science and technology and to the wide application in areas such as electrorheological fluids (smart fluids) which have the property that the viscosity changes drastically when exposed to heat or electrical fields. To tackle and understand these models, new sophisticated mathematical functional spaces have been introduced, such as the Lebesgue and Sobolev spaces with variable exponents. In this work, we are concerned with a system of wave equations with variable-exponent nonlinearities. This system can be regarded as a model for interaction between two fields describing the motion of two "smart" materials. We, first, establish the existence of global solutions then show that solutions of enough regularities stabilize to the rest state (0, 0) either exponentially or polynomially depending on the range of the variable exponents. We also present some numerical tests to illustrate our theoretical findings.