Abstract We study the long-time behavior of the solution to a type of dissipative wave equation, where the operator in the equation is time-dependent and the solution is defined on a metric measure space ( X , m ) satisfying appropriate conditions. The operator is assumed to be self-adjoint and is related to a time-dependent Dirichlet form. We link hyperbolic PDEs with the firmly established theories for parabolic PDEs in metric measure spaces and Dirichlet forms, subsequently deriving the asymptotic behavior of the solution to the dissipative wave equation. We present several nontrivial examples of dissipative wave equations in metric measure spaces where our theory works.

中文翻译：

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度量空间中耗散波动方程的扩散现象

摘要 我们研究了一类耗散波动方程解的长期行为，其中方程中的算子是时间相关的，并且该解是在满足适当条件的度量测度空间 ( X , m ) 上定义的。假设算子是自伴随的，并且与时间相关的狄利克雷形式有关。我们将双曲偏微分方程与公制测量空间和狄利克雷形式中的抛物线偏微分方程的牢固建立的理论联系起来，随后推导出耗散波动方程解的渐近行为。我们在我们的理论适用的度量测量空间中展示了耗散波动方程的几个重要示例。