Deterministic and random attractors for a wave equation with sign changing damping
Deterministic and random attractors for a wave equation with sign changing damping
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DOI：
10.4213/im9250e
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
Qingquan Chang;Dandan Li;Chunyou Sun;S. Zelik
中科院分区：
文献类型：
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作者：
Qingquan Chang;Dandan Li;Chunyou Sun;S. Zelik
本文详细研究了三维有界区域中弱阻尼波动方程产生的长时间动力学问题，其中阻尼系数显式地依赖于时间，并且可以改变符号。结果表明，在这种情况下，其中的非线性是超线性的，所考虑的方程仍然是耗散的，如果耗散率的加权平均值仍然是积极的，这种类型的条件是不充分的线性情况下。两个主要不同的情况下考虑。在这种情况下，当这个平均值是均匀的（这对应于确定性耗散率），它表明，所考虑的系统具有光滑的均匀吸引子以及非自治指数吸引子。在平均值不均匀的情况下（这对应于随机耗散率，例如，当该耗散率由伯努利过程生成时），构造了回火随机吸引子。与通常的情况相反，这种随机吸引子预计具有无限的Hausdorff维数和分形维数。最后给出了随机吸引子无穷维性的简化模型实例。
The paper gives a detailed study of long-time dynamics generated by weakly damped wave equations in bounded 3D domains where the damping coefficient depends explicitly on time and may change sign. It is shown that in the case, where the non-linearity is superlinear, the considered equation remains dissipative if the weighted mean value of the dissipation rate remains positive and that the conditions of this type are not sufficient in the linear case. Two principally different cases are considered. In the case when this mean is uniform (which corresponds to deterministic dissipation rate), it is shown that the considered system possesses smooth uniform attractors as well as non-autonomous exponential attractors. In the case where the mean is not uniform (which corresponds to the random dissipation rate, for instance, when this dissipation rate is generated by the Bernoulli process), the tempered random attractor is constructed. In contrast to the usual situation, this random attractor is expected to have infinite Hausdorff and fractal dimensions. The simplified model example demonstrating infinite-dimensionality of the random attractor is also presented.