Decay of the 3 D Quasilinear Hyperbolic Equations with Nonlinear Damping

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 20K. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in the sense of H3-norm. Furthermore, if, additionally, Lp-norm (1 ≤ p < 6/5) of the initial perturbation is finite, we also prove the optimal Lp-L2 decay rates for such a solution without the additional technical assumptions for the nonlinear damping f(V) given by Li and Saxton.


Introduction
In this paper, we consider the following 3D quasilinear hyperbolic system with nonlinear damping: V  − div (ℎ (V) P) = 0,  ∈ R 3 ,  > 0, P  + ∇ (V) =  (V) P,  ∈ R 3 ,  > 0, (1) with initial data (V, P) (, 0) = (V 0 , P 0 ) () → (V, 0) , where   (V) < 0, ℎ(V) > 0, (V) < 0, and V > 0. 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  ()  + div q = 0, where  > 0 is the absolute temperature,  is the internal energy, and q is the heat flux.The second equation in (1) is the evolution equation for an internal parameter P, 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, −(−) ∇ (x, ) ,  () > 0,  > 0. (4) By defining then ( 4) can be equivalently replaced with q = − () P, Equation (7), related to (4) via (5), is however linear and does not fully describe the properties of heat propagation in solids (cf.[1][2][3][4] and references therein).To improve the model, one may generalize the history dependence of q by modifying (4) or, as was done in [2], by introducing a suitable nonlinear dependence in (7), P  =  1 () ∇ +  2 () P. (8)           F) , for  ∈ R. (12) We will use the notation  ≲  to mean that  ≤  for a generic constant  > 0 which may vary at different places.Now, we are in a position to state our main results.
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][6][7], the approach is new and quite different here.In [5][6][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 onedimensional problem.The main difficulties in this step of the paper arise from the terms involving ∇ +1 V or ∇  div P 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   -L 2 decay rates for the linearized system, we can prove the optimal   - 2 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   - 2 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   - 2 decay estimates obtained in Section 3 and the uniform nonlinear estimates obtained in Section 4.

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 (V, P).
As usual, Theorem 1 will be proved by combining the local existence result together with uniform a priori energy estimates.
Assume that the Cauchy problem (18) has a solution (V, P)(, ) in the same function class as in Proposition 2 on where  > 0 is a positive constant.Then there exists a small positive constant , which is independent of , such that if then, for any  ∈ [0, ], it holds that Furthermore, if additionally ‖V 0 −V‖   +‖P 0 ‖  3/(3−) is bounded for some 1 ≤  < 6/5, then, for any  ∈ [0, ], the following decay estimates of the solution (V, P) 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.

𝐿 𝑃 -𝐿 2 Decay Estimates for the Linearized Equations
The corresponding linearized system to (18) is V  +  1 div P = 0, Due to the fact that the Fourier transform of ( be the "compressible part" of P, and let be the "incompressible part" of P (with (curl ) ), 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  3 (R 3 ) for the velocity P are the same as for (, ).
In the rest of this section, we devote ourselves to show the following   - 2 decay estimates on the linearized system (33).

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  > 0

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We first derive the following lower order energy estimate of the solutions.Lemma 5.There exists a suitably large constant  1 > 0, which is independent of , such that for any 0 ≤  ≤ .
Proof.Multiplying (18) 1 -(18) 2 by V, P, respectively, and then integrating them over R 3 , 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 (V) < 0 and  > 0 is suitably small.Next we shall estimate the term ‖∇V‖ 2 .Multiplying (18) 2 by ∇V and then integrating it over, we obtain where from (18) 1 , we can rewrite the first term in the righthand 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 ⟨P, ⟩, it also holds that Putting (63) into (62) and noting that  > 0 is sufficiently small, we obtain Finally, multiplying (58) by a sufficiently large positive constant  1 and then adding it to (64), we can get (54) since  > 0 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  2 > 0, which is independent of , such that for any 0 ≤  ≤ , where  = ℎ(V + V) − ℎ(V).(66)

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Next, we will estimate the two terms in the right-hand side of (66).To do this, by the definition of , we get (67) We will estimate the four terms   1 with 1 ≤  ≤ 4. The main difficulties arise from terms involving ∇ +1 V or ∇  div P 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 ,  1  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  1,3 1 .If  = 1, from integration by parts, Lemmas A.1 and A.4, we obtain

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If  = 2, by integration by parts, Hölder inequality, and Lemmas A.1 and A.4, we have If  = 3, 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 and ( 74)-( 75) into (68) leads to Similarly, for the terms Using the similar arguments in obtaining (74), for the term  3 1 , it also holds that Putting ( 76)-( 78) into (67) leads to Next, we estimate the term  2 .By virtue of the definition of , we separate  2 into three parts Putting the estimates (81)-( 82) into (80) gives Substituting ( 79) and ( 83) into (66) and then summing up the resultant equation for  = 1 to 3, we obtain Finally, we derive the estimate on ‖∇∇  V‖ 2 for 1 ≤  ≤ 2. Applying ∇  V to (18) 2 , multiplying the resulting identity by ∇∇  V, and then integrating over R 3 , we obtain where from (18) 1 , we can rewrite the first term in the righthand side of (85) as Similarly, for ⟨∇∇  P, ∇  ⟩, it also holds that Putting (89) into (88) and summing up for  = 1 to 2, we obtain Therefore, multiplying (84) by a sufficiently large positive constant  2 and adding it to (90), we can obtain (65) since  > 0 is small enough.Therefore, we have completed the proof of Lemma 6.

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 0 ≤  ≤ , where is finite due to the assumptions of Proposition 3.
Step 1.Since  > 0 is suitably small, from Lemmas 5 and 6, we can choose a suitably large positive constant  3 such that Due to (53), it is clear that the expression under / in (94) is equivalent to ‖(V, P)()‖ 2 3 .Hence, by integrating (94) directly in time, we obtain (24).