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We investigate the 3D quasilinear hyperbolic equations with nonlinear damping which describes the propagation of heat wave for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in the sense of

In this paper, we consider the following 3D quasilinear hyperbolic system with nonlinear damping:

Finally, by employing the substitution

To go directly to the theme of this paper, we now only review some former results closely related. For the one-dimensional version of model (

To sum up, it is still unknown on the decay rates of solutions to the 3D quasilinear hyperbolic system with nonlinear damping (

Before stating the main results, we introduce some notations for the use throughout this paper. We use

Now, we are in a position to state our main results.

Assume that

We now comment on the proof of Theorem

Firstly, we deduce the optimal _{1}-(_{2} has multiple eigenvalue and the semigroup theory, to investigate spectral structure of (_{1}-(_{2}, we need to analyze the corresponding Jordan structure. Our main idea is to introduce Hodge decomposition to overcome this difficulty. By applying Hodge decomposition to system (_{1}-(_{2}, we can transform system (_{1}-(_{2} into two systems. One only has one equation, and another has two different eigenvalues.

Secondly, we establish the uniform energy estimates to the original system (_{1}-(_{2}. Based on the system’s special dissipation structure and delicate analysis and interpolation technique on the nonlinear terms, the desired energy estimates can be achieved. Compared to the one-dimensional results in [_{1}-(_{2}, the technique on interpolation and energy estimates, and a technical lemma on estimating the spatial derivatives of nonlinear function to tackle these difficulties. It is worth mentioning that, in our proofs, we do not need the technical assumptions (

Finally, we deduce the decay rates on the solutions. By virtue of the results obtained in the first two steps and by virtue of the uniform nonlinear energy estimates and the optimal

The rest of the paper is organized as follows. In Section

Denoting

As usual, Theorem

Let

Let

Theorem

The corresponding linearized system to (_{1}-(_{2} has multiple eigenvalue and the semigroup theory, to analyze spectral structure of (_{1}-(_{2}, we need to consider corresponding Jordan structure. In spirit of [_{1}-(_{2}, we can transform system (_{1}-(_{2} into two systems. One only has one equation, and another has two different eigenvalues. To begin with, let_{1}-(_{2} as follows:

In the rest of this section, we devote ourselves to show the following

Assume that _{1}-(_{2}, and let

Since the proof of (_{1}-(_{2} has the following expression:_{1}-(_{2}. Next, we derive the explicit expression for the Fourier transform _{1}-(_{2}, we obtain

In order to deduce the long-time decay estimates of solutions in

Therefore, we have proved Proposition

In this section, we deduce the uniform a priori estimates stated in Proposition

We first derive the following lower order energy estimate of the solutions.

There exists a suitably large constant

Multiplying (_{1}-(_{2} by

First, for the first term, by virtue of Hölder inequality, Cauchy inequality, and Lemmas

Next we shall estimate the term _{2} by _{1}, we can rewrite the first term in the right-hand side as

In the following lemma, we deduce the higher order energy estimate of the solutions.

There exists a suitably large constant

For _{1}-(_{2} and multiplying the resulting identities by _{1}-(_{2}, the technique on interpolation and energy estimates, and a technical lemma on estimating the spatial derivatives of nonlinear function to tackle these difficulties. It should be mentioned that, in our proofs, we do not need the technical assumptions (_{1}, _{1} and (_{1}, (

Next, we estimate the term

Finally, we derive the estimate on _{2}, multiplying the resulting identity by _{1}, we can rewrite the first term in the right-hand side of (

In this section, we devote ourselves to prove Proposition

It holds that

Applying (

Now, we are in a position to prove Proposition

The proof involves the following two steps.

Due to Lemma

Next, we turn to the proofs of (

Finally, we prove (

We will extensively use the Sobolev interpolation of the Gagliardo-Nirenberg inequality.

Let

This is a special case of [

Next, we recall the following product estimate.

Let

For

We also need the following commutator estimate.

Let

The proof can be found in [

Finally, to estimate the

Assume that

Notice that, for

The authors declare that there are no conflicts of interest regarding the publication of this article.

This work was partially supported by the Hunan Provincial Natural Science Foundation of China no. 2017JJ2105, the National Natural Science Foundation of China no. 11571280, no. 11301172, and no. 11226170, and National Scholarship Fund in Hunan Province Cooperation Projects.

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