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Ultrasonic propagation through media with thermal and molecular relaxationcan be modeled by third-order in time nonlinear wave-like equations withmemory. This paper investigates the asymptotic behavior of a Cauchy problem forsuch a model, the nonlocal Jordan--Moore--Gibson--Thompson equation, in theso-called critical case, which corresponds to propagation in inviscid fluids.The memory has an exponentially fading character and type I, meaning thatinvolves only the acoustic velocity potential. A major challenge in the globalanalysis is that the linearized equation's decay estimates are ofregularity-loss type. As a result, the classical energy methods fail to workfor the nonlinear problem. To overcome this difficulty, we constructappropriate time-weighted norms, where weights can have negative exponents.These problem-tailored norms create artificial damping terms that help controlthe nonlinearity and the loss of derivatives, and ultimately allow us todiscover the model's asymptotic behavior.

Asymptotic behavior of nonlinear sound waves in inviscid media with thermal and molecular relaxation