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In this paper, we consider a coupled system of Laplacian and bi-Laplacian equations with nonlinear dampings and source terms of variable-exponents nonlinearities. This system is supplemented with initial and mixed boundary conditions. First, we establish the existence and uniqueness results of a weak solution, under suitable assumptions on the variable exponents. Second, we show that the solutions with positive-initial energy blow-up in a finite time. Finally, we establish the global existence as well as the energy decay results of the solutions, using the stable-set and the multiplier methods, under appropriate conditions on the variable exponents and the initial data. [ABSTRACT FROM AUTHOR]

In this work, we consider a swelling porous system where the damping terms are on the boundary. We establish an explicit and general decay result, without imposing restrictive growth assumption near the origin on the damping terms. Our result allows a larger class of damping terms, and the usual exponential and polynomial decay estimates are only special cases. We also give some illustrative examples. [ABSTRACT FROM AUTHOR]

With the advancement of science and technology, many physical and engineering models require more sophisticated mathematical functional spaces to be studied and well understood. For example, in fluid dynamics, electrorheological fluids (smart fluids) have the property that the viscosity changes (often drastically) when exposed to an electrical field. The Lebesgue and Sobolev spaces with variable exponents proved to be efficient tools to study such problems as well as other models like the image processing. In this work, we consider the following nonlinear wave equation with variable exponents: utt − Δu-Δut + | ut |m ( ⋅ )-2 ut = 0, in Ω × (0,T), where Ω is a bounded domain and T > 0, and show that weak solutions decay exponentially or polynomially depending on the range of the variable exponent m . We also give two numerical examples to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]