Global well-posedness of the Cauchy problem for the Jordan--Moore--Gibson--Thompson equation with arbitrarily large higher-order Sobolev norms

Abstract

In this paper, we consider the 3D Jordan-Moore-Gibson-Thompson equation arising in nonlinear acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This improves some available results in the literature. Second, we prove a new decay estimate for the linearized model and removing the L 1 -assumption on the initial data. The proof of this decay estimate is based on the high-frequency and low-frequency decomposition of the solution together with an interpolation inequality related to Sobolev spaces with negative order.

On the Cauchy problem for the Jordan-Moore-Gibson-Thompson equation in the dissipative case

“…Let us turn to the JMGT equation (3), which has been studied by [18,19,24,3,25] and references therein. Additionally, the semilinear MGT equation in the conservative case for R n has been investigated by [5,6].…”

“…Concerning the Cauchy problem for (3) in the dissipative case, [24] demonstrated global (in time) well-posedness result in 3D by using higher-order energy methods together with a bootstrap argument, and derived some decay estimates of solutions with weighted L 1 assumption on initial data. Quite recently, the author of [25] improved the global (in time) existence result for 3D, where the smallness assumption for higher-order regular data was removed, whose idea is to utilize negative Sobolev spaces. To the best of the author's knowledge, the application of energy method to prove global (in time) existence of solutions required some suitable constructions of energy.…”

“…For the sake of simplicity, we denote ψ I,j k (x) := ∂ k t ψ I,j (0, x) for k = 0, 1, 2. Then, concerning j ∈ Z now, we can get from (25) with the terms behaving as τ j+1 that…”

Asymptotic behaviors for the Jordan-Moore-Gibson-Thompson equation in the viscous case

“…The authors of [28] demonstrated global (in time) well-posedness result in three-dimensions by higher-order energy methods together with a bootstrap argument, and derived some decay estimates of solutions. Quite recently, the author of [29] improved the global (in time) existence result in [28] for three-dimensions, where the smallness assumption for higher-order regular data was dropped in the framework of Sobolev spaces with negative index. So far the sharp estimates and asymptotic profiles of solutions are still open, especially, in the physical dimensions n = 1, 2, 3.…”

On the Cauchy problem for acoustic waves in hereditary fluids: decay properties and inviscid limits

“…Concerning the MGT equation and the JMGT equation, some qualitative properties of solutions (well-posedness, stabilities, decay estimates, singular limits, inviscid limits and asymptotic profiles) have been deeply analyzed. We refer interested readers to [28,29,36,11,42,2,30,43,3,9,10,8,7,31,44] and references therein.…”