Nonlinear sound propagation through media with thermal and molecular relaxation can be modeled by third-order in time wave-like equations with memory. We investigate the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan–Moore–Gibson–Thompson equation, in the so-called critical case, which corresponds to propagation through inviscid fluids or gases. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in studying global behavior is that the linearized equation’s decay estimates are of regularity-loss type. As a result, the classical energy methods fail to work for the nonlinear problem. To overcome this difficulty, we construct appropriate time-weighted norms, where weights can have negative exponents. These problem-tailored norms create artificial damping terms that help control the nonlinearity and the loss of derivatives, and ultimately allow us to discover the model’s asymptotic behavior.

通过具有热和分子弛豫的介质的非线性声音传播可以通过具有记忆的时间波状方程中的三阶建模。我们研究了这种模型的柯西问题的渐近行为，即非局部 Jordan-Moore-Gibson-Thompson 方程，在所谓的临界情况下，对应于通过无粘性流体或气体的传播。记忆具有指数衰减特征和类型 I，这意味着仅涉及声速势。研究全局行为的一个主要挑战是线性化方程的衰减估计是规律性损失类型。因此，经典的能量方法无法解决非线性问题。为了克服这个困难，我们构建了适当的时间加权规范，其中权重可以具有负指数。