Advances in Mathematical Physics
Volume 2017 (2017), Article ID 2708483, 13 pages
https://doi.org/10.1155/2017/2708483

Research Article

Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

Hongjun Qiu¹ and Yinghui Zhang²

¹College of Mathematics and Computer Science, Hunan Normal University, Changsha 410081, China
²Department of Mathematics, Hunan Institute of Science and Technology, Yueyang 414006, China

Correspondence should be addressed to Yinghui Zhang; zhangyinghui0910@126.com

Received 13 February 2017; Accepted 19 April 2017; Published 23 May 2017

Academic Editor: Ming Mei

Copyright © 2017 Hongjun Qiu and Yinghui Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 -norm. Furthermore, if, additionally, -norm () of the initial perturbation is finite, we also prove the optimal decay rates for such a solution without the additional technical assumptions for the nonlinear damping given by Li and Saxton.

1. Introduction

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

with initial data

where , , , and . The above system is derived in [1, 2] and describes the propagation of heat wave for rigid solids at very low temperatures, below about 20 K. The first equation in (1) comes from the balance of energy, which takes the form

where is the absolute temperature, is the internal energy, and is the heat flux. The second equation in (1) is the evolution equation for an internal parameter , which is introduced to account for memory effects of the heat flux. The effect of memory may be considered, for example, as a functional of a history of temperature gradient,

By defining

then (4) can be equivalently replaced with

Equation (7), related to (4) via (5), is however linear and does not fully describe the properties of heat propagation in solids (cf. [1–4] and references therein). To improve the model, one may generalize the history dependence of by modifying (4) or, as was done in [2], by introducing a suitable nonlinear dependence in (7),

The functions , , and present in (6) and (8) are material functions. The second law of thermodynamics imposes the restrictions that and , where the constant comes from the Helmholtz free energy which has the form . We additionally make an assumption that (cf. [2]). Combining (3) with (8) gives the following system:

Finally, by employing the substitution with , , and , system (9) is exactly system (1).

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 (1),

the existence and asymptotic behavior of smooth solutions for the Cauchy problem and initial boundary value problem have been considered by [5, 6] and [7], respectively. In [5], they obtained the convergence rates of the smooth solutions for the Cauchy problem under the following technical assumptions for the nonlinear damping:

The authors in [6] removed restriction (11). Recently, the authors in [7] proved convergence rates for the half space problem. Moreover, the authors in [3] and [4] considered phase transition and the effect of damping respectively. When with being a given positive constant and , system (10) reduces to the well-known -system with linear damping. The investigation of the case of -system with linear damping has been extensively studied. We refer reader to [8–46] and references therein.

To sum up, it is still unknown on the decay rates of solutions to the 3D quasilinear hyperbolic system with nonlinear damping (1)-(2). In this paper, we will give a positive answer to this question. More precisely, we prove global existence and uniqueness of strong solutions when the initial data is near its equilibrium in the sense of -norm. Moreover, if, additionally, -norm () of the initial perturbation is finite, we also show the optimal decay rates for the solutions without the additional technical assumptions for the nonlinear damping given by Li and Saxton in [5].

Before stating the main results, we introduce some notations for the use throughout this paper. We use and to denote the usual Sobolev spaces with norm and , respectively, for and . In particular, for , we will use and for simplicity. We use to denote the inner-product in and to denote the pseudo differential operator:

We will use the notation to mean that for a generic constant which may vary at different places.

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

Theorem 1. Assume that ,  ,  ,  ,  ,  ,  , and is small enough. Then the Cauchy problem (1)-(2) has a unique global solution with , which satisfies

Furthermore, for any , the following energy estimates hold:

Finally, if further is bounded for some , then the following decay estimates of the solution hold:

We now comment on the proof of Theorem 1. Roughly speaking, we follow the framework of [47, 48] on two-phase fluid model, and the proof consists of following three steps.

Firstly, we deduce the optimal decay rates on the solutions to the corresponding linearized system. By making delicate pointwise analysis on the linearized system, we can prove that the variables and have the decay rates and , respectively, when is finite with . Due to the fact that the Fourier transform of (30)₁-(30)₂ has multiple eigenvalue and the semigroup theory, to investigate spectral structure of (30)₁-(30)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 (30)₁-(30)₂, we can transform system (30)₁-(30)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 (18)₁-(18)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 [5–7], the approach is new and quite different here. In [5–7], the antiderivative technique plays an essential role in proving their main results. However, the antiderivative technique does not work in our high dimensional problem since the antiderivative technique is basically limited to one-dimensional problem. The main difficulties in this step of the paper arise from the terms involving or which are not included in left hand side of (65) (see Lemma 6 for more details). We use the special dissipation structure of (18)₁-(18)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 (11) as in [5].

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 decay rates for the linearized system, we can prove the optimal decay rates for the solutions.

The rest of the paper is organized as follows. In Section 2, we first reformulate the problem and then prove Theorem 1. In Sections 3 and 4, we derive the optimal decay estimates for the linearized system and the uniform nonlinear estimates for the original equations, respectively. In Section 5, we prove Proposition 3 by combining the optimal decay estimates obtained in Section 3 and the uniform nonlinear estimates obtained in Section 4.

2. Reformulation and the Proof of Theorem 1

Denoting

and making change of variables by

we can reformulate the Cauchy problem (1)-(2) as

where

Here and in the sequel, for the notational simplicity, we will denote the reformulated variables by .

As usual, Theorem 1 will be proved by combining the local existence result together with uniform a priori energy estimates.

Proposition 2 (local existence). Let be such that

Then there exists a positive constant depending on such that the unique solution of the Cauchy problem (18) exists on and satisfies

Furthermore, one has the following estimates:

Proposition 3 (a priori estimate). Let . Assume that the Cauchy problem (18) has a solution in the same function class as in Proposition 2 on , where is a positive constant. Then there exists a small positive constant , which is independent of , such that if

then, for any , it holds that

Furthermore, if additionally is bounded for some , then, for any , the following decay estimates of the solution hold:

Proof of Theorem 1. Theorem 1 follows from Propositions 2 and 3 by the standard continuity argument. The proof of Proposition 2 is standard whose proof can be found in [49, 50]. Proposition 3 will be proved in Section 5.

3. Decay Estimates for the Linearized Equations

The corresponding linearized system to (18) is

Due to the fact that the Fourier transform of (30)₁-(30)₂ has multiple eigenvalue and the semigroup theory, to analyze spectral structure of (30)₁-(30)₂, we need to consider corresponding Jordan structure. In spirit of [40, 47, 48], we will use Hodge decomposition technique to overcome this difficulty. By applying Hodge decomposition to system (30)₁-(30)₂, we can transform system (30)₁-(30)₂ into two systems. One only has one equation, and another has two different eigenvalues. To begin with, let

be the “compressible part” of , and let

be the “incompressible part” of (with ), then we can rewrite system (30)₁-(30)₂ as follows:

Noticing the definitions of and and the relation

which involves pseudo differential operators with zero degree, one can easily see that the estimates in space for the velocity are the same as for .

In the rest of this section, we devote ourselves to show the following decay estimates on the linearized system (33).

Proposition 4. Assume that and is bounded with . Denote by the solution of (30)₁-(30)₂, and let and . Then, for , one has

Proof. Since the proof of (35) is trivial, we will focus on the proofs of (36) and (37). Firstly, in terms of semigroup theory, the solution of system (33)₁-(33)₂ has the following expression:

where is Green’s matrix of system (33)₁-(33)₂. Next, we derive the explicit expression for the Fourier transform of Green’s matrix . Taking the Fourier transform to (33)₁-(33)₂, we obtain

From (39) and a simple calculation, we have

By a straightforward computation, we can get the expression of the solution to the ODE (40)

where are the eigenvalues of the ODE (40). To compute , we substitute (41) into (39) to get

Thus, from (41)-(42), we obtain the explicit expression of the Fourier transformation of Green’s matrix as

In order to deduce the long-time decay estimates of solutions in -framework, we need to verify the approximation of . By virtue of the definition , for , it holds that

Similarly, for high frequency, it also holds that

where is some given positive constant and

By virtue of formula (43) and the asymptotical estimates on its elements, we are in a position to prove (36) and (37). By virtue of (43)–(45), (47)-(48), (50), Parsevel’s identity, Hausdorff-Young’s inequality, and Hölder inequality, we deduce that, for each ,

This proves (36). Similarly, for each , we also have

which together with (34)-(35) yields (37).
Therefore, we have proved Proposition 4.

4. Uniform a Priori Estimates

In this section, we deduce the uniform a priori estimates stated in Proposition 3. Throughout this section, we assume that all the conditions in Proposition 3 are satisfied. Furthermore, we assume a priori that for sufficiently small

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

Lemma 5. There exists a suitably large constant , which is independent of , such that

for any .

Proof. Multiplying (18)₁-(18)₂ by , , respectively, and then integrating them over , we have

We will estimate the two terms in the right-hand side of (55) as follows.
First, for the first term, by virtue of Hölder inequality, Cauchy inequality, and Lemmas A.1 and A.4, we get

Similarly, for the second term, it also holds that

Thus substituting (56) and (57) into (55) gives

since and is suitably small.
Next we shall estimate the term . Multiplying (18)₂ by and then integrating it over, we obtain

where from (18)₁, we can rewrite the first term in the right-hand side as

By virtue of the definition of , we obtain

where Lemma A.4 has been used. Then, by combining the relations (59)–(61), we have

Similar to the proof of the estimate on , it also holds that

Putting (63) into (62) and noting that is sufficiently small, we obtain

Finally, multiplying (58) by a sufficiently large positive constant and then adding it to (64), we can get (54) since is small enough. Therefore, we have completed the proof of Lemma 5.

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

Lemma 6. There exists a suitably large constant , which is independent of , such that

for any , where .

Proof. For , by applying to (18)₁-(18)₂ and multiplying the resulting identities by , , respectively, summing up them, and then integrating over , we have

Next, we will estimate the two terms in the right-hand side of (66). To do this, by the definition of , we get

We will estimate the four terms with . The main difficulties arise from terms involving or which are not included in left hand side of (65). The main observation is that we can tackle these difficulties by using the special dissipation structure of (18)₁-(18)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 (11) as in [5]. Firstly, by virtue of (18)₁, can be rewritten as

From (18)₁ and (53), we obtain

By virtue of (18)₁, (53), Lemmas A.1–A.4, Cauchy inequality, and Hölder inequality, we get

Next, we estimate the term . If , from integration by parts, Lemmas A.1 and A.4, we obtain

If , by integration by parts, Hölder inequality, and Lemmas A.1 and A.4, we have

If , by using similar arguments as the above, we obtain

Combining (71)–(73) gives that

From (53), (56), Lemmas A.1–A.4, Cauchy inequality, and Hölder inequality, we get

Putting (69)-(70) and (74)-(75) into (68) leads to

Similarly, for the terms and , we also have

Using the similar arguments in obtaining (74), for the term , it also holds that

Putting (76)–(78) into (67) leads to

Next, we estimate the term . By virtue of the definition of , we separate into three parts

From (53), (56), Lemmas A.1–A.4, Cauchy inequality, and Hölder inequality, we get

Using similar arguments in obtaining (76), for , it also holds that

Putting the estimates (81)-(82) into (80) gives

Substituting (79) and (83) into (66) and then summing up the resultant equation for to , we obtain

Finally, we derive the estimate on for . Applying to (18)₂, multiplying the resulting identity by , and then integrating over , we obtain

where from (18)₁, we can rewrite the first term in the right-hand side of (85) as

By virtue of the definition of , (53), (56), and Lemmas A.1–A.4, we get

Then, combining relations (85)–(87) yields

Similarly, for , it also holds that

Putting (89) into (88) and summing up for to , we obtain

Therefore, multiplying (84) by a sufficiently large positive constant and adding it to (90), we can obtain (65) since is small enough. Therefore, we have completed the proof of Lemma 6.

5. Global Existence and Decay Rates

In this section, we devote ourselves to prove Proposition 3. To begin with, we give the following lemma which will be used later.

Lemma 7. It holds that

for any , where is finite due to the assumptions of Proposition 3.

Proof. Applying (36)-(37) with and using Duhamel’s principle, we have

From (53), (56), Lemmas A.1 and A.4, and Hölder inequality, we can estimate the nonlinear source terms as follows:

Substituting these estimates into (92) gives (91) and thus proves Lemma 7.

Now, we are in a position to prove Proposition 3.

Proof of Proposition 3. The proof involves the following two steps.
Step  1. Since is suitably small, from Lemmas 5 and 6, we can choose a suitably large positive constant such that

Due to (53), it is clear that the expression under in (94) is equivalent to . Hence, by integrating (94) directly in time, we obtain (24).
Step  2. Define the temporal energy functional

for any , where is equivalent to since can be large enough.
Due to Lemma 6, we have

Adding to both sides of (96) gives

where is a positive constant independent of . Setting

then

This together with (91) yields that

Thus, from Gronwall’s inequality, (97), and (100), we have

Since is nondecreasing, from (101), we have

for any , which together with the smallness of gives that

Therefore, (99) and (103) lead to

and this proves (28).
Next, we turn to the proofs of (25)–(27). Applying (36)-(37) with , we have from Duhamel’s principle that

for any . On the other hand, from Lemma A.1 and (29), we also have

Thus, we can deduce (25)–(27) from the interpolation and (105)-(106) immediately.
Finally, we prove (29). By virtue of (18) and (28), we have

for any . Therefore, (29) is proved and therefore we have completed the proof of Proposition 3.

Appendix

Analytic Tools

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

Lemma A.1. Let ; then one has

where satisfies

Proof. This is a special case of [33, pp. 125, Theorem].

Next, we recall the following product estimate.

Lemma A.2. Let be an integer; then it holds that

where and

Proof. For , it can be proved by using Lemma A.1. For the general cases, one may refer to [51, Lemma  3.1].

We also need the following commutator estimate.

Lemma A.3. Let be an integer and define the commutator

Then one has

where and

Proof. The proof can be found in [51, Lemma  3.1].

Finally, to estimate the -norm of the spatial derivatives of ,  , and , we shall need the technique lemma.

Lemma A.4. Assume that . Let be a smooth function of with bounded derivatives of any order, then, for any integer and , one has

Proof. Notice that, for ,

where the functions are some derivatives of and ,  , with . We then use the Sobolev interpolation of Lemma A.1 to bound

Hence, we conclude our lemma since .

Conflicts of Interest

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

Acknowledgments

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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