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.

中文翻译：

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变指数非线性波系统的存在性和稳定性

具有可变指数的问题最近引起了广泛的关注，并且已经建立了各种存在，不存在和稳定的结果。由于科学和技术的最新发展以及在诸如电流变流体（智能流体）之类的领域中的广泛应用，已经表明了这些问题的重要性，所述领域具有当暴露于热或电场时粘度急剧变化的性质。为了处理和理解这些模型，引入了新的复杂数学函数空间，例如具有可变指数的Lebesgue和Sobolev空间。在这项工作中，我们关注具有可变指数非线性的波动方程组。该系统可以看作是描述两个“智能”物料运动的两个场之间相互作用的模型。我们，首先，建立全局解的存在性，然后证明具有足够规律性的解根据变量指数的范围呈指数或多项式稳定到静止状态（0,0）。我们还提出了一些数值测试来说明我们的理论发现。