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Advances in Mathematical Physics
Volume 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

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; moc.621@0190iuhgniygnahz

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 datawhere , , , 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 formwhere 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 definingthen (4) can be equivalently replaced withEquation (7), related to (4) via (5), is however linear and does not fully describe the properties of heat propagation in solids (cf. [14] 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 [846] 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 satisfiesFurthermore, 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 [57], the approach is new and quite different here. In [57], 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

Denotingand making change of variables bywe can reformulate the Cauchy problem (1)-(2) aswhereHere 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 thatThen there exists a positive constant depending on such that the unique solution of the Cauchy problem (18) exists on and satisfiesFurthermore, 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 ifthen, for any , it holds thatFurthermore, 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) isDue 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, letbe the “compressible part” of , and letbe the “incompressible part” of (with ), then we can rewrite system (30)1-(30)2 as follows:Noticing the definitions of and and the relationwhich 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 obtainFrom (39) and a simple calculation, we haveBy 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 getThus, from (41)-(42), we obtain the explicit expression of the Fourier transformation of Green’s matrix asIn 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 thatSimilarly, for high frequency, it also holds thatwhere is some given positive constant andBy 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 havewhich 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 thatfor any .

Proof. Multiplying (18)1-(18)2 by , , respectively, and then integrating them over , we haveWe 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 getSimilarly, for the second term, it also holds thatThus substituting (56) and (57) into (55) givessince and is suitably small.
Next we shall estimate the term . Multiplying (18)2 by and then integrating it over, we obtainwhere from (18)1, we can rewrite the first term in the right-hand side asBy virtue of the definition of , we obtainwhere Lemma A.4 has been used. Then, by combining the relations (59)–(61), we haveSimilar to the proof of the estimate on , it also holds thatPutting (63) into (62) and noting that is sufficiently small, we obtainFinally, 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 thatfor 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 haveNext, we will estimate the two terms in the right-hand side of (66). To do this, by the definition of , we getWe 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 asFrom (18)1 and (53), we obtainBy virtue of (18)1, (53), Lemmas A.1A.4, Cauchy inequality, and Hölder inequality, we getNext, we estimate the term . If , from integration by parts, Lemmas A.1 and A.4, we obtainIf , by integration by parts, Hölder inequality, and Lemmas A.1 and A.4, we haveIf , by using similar arguments as the above, we obtainCombining (71)–(73) gives thatFrom (53), (56), Lemmas A.1A.4, Cauchy inequality, and Hölder inequality, we getPutting (69)-(70) and (74)-(75) into (68) leads toSimilarly, for the terms and , we also haveUsing the similar arguments in obtaining (74), for the term , it also holds thatPutting (76)–(78) into (67) leads toNext, we estimate the term . By virtue of the definition of , we separate into three partsFrom (53), (56), Lemmas A.1A.4, Cauchy inequality, and Hölder inequality, we getUsing similar arguments in obtaining (76), for , it also holds thatPutting the estimates (81)-(82) into (80) givesSubstituting (79) and (83) into (66) and then summing up the resultant equation for to , we obtain