The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy

Yanbo Hu; Yuusuke Sugiyama

doi:10.1515/ans-2023-0142

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The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy

Yanbo Hu und Yuusuke Sugiyama

Veröffentlicht/Copyright: 17. Juni 2024

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Aus der Zeitschrift Advanced Nonlinear Studies Band 24 Heft 4

Abstract

We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time T of the solution is independent of the spatial variable.

Keywords: thermoelasticity system; well-posedness; classical solvability; degeneracy

MSC 2020: Primary 35A01; Secondary 35L20; 35M11; 35L15

1 Introduction

In this paper, we are interested in the Cauchy problem to the one-dimensional nonlinear hyperbolic-parabolic coupled system

(1) u t t − u 2 a u x x = θ x x , θ t − θ x x = f ( θ ) u t ,

supplemented with the initial conditions

(2) u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) , θ ( 0 , x ) = θ 0 ( x ) .

Here f(θ) is a given smooth function and a > 0 is a constant.

System (1) can be derived from the thermoelasticity theory, which incorporates the effects of displacement and temperature of the elastic material. In the absence of external heat source and force, the Lagrangian formulation of the laws of balance of mass, momentum, and energy can be written as [1], [2], [3], [4]

(3) u t − v x = 0 , v t − σ x = 0 , e + v 2 2 t − ( σ v ) x = q x ,

where u, v, σ, e and q are the strain, the velocity, the stress, the internal energy and the heat flux, respectively. The constitutive relations of a thermoelastic material take the forms

σ = σ ( u , T ) , e = e ( u , T ) , q = q ( u , T , T x ) ,

where T is the absolute temperature, and T _x expresses the derivative of T with respect to x. Moreover, σ and e are required to satisfy the compatibility relation

(4) e u ( u , T ) = σ ( u , T ) − T σ T ( u , T ) , e T ( u , T ) > 0 ,

which arises from the second law of thermodynamics. Furthermore, the heat flux q must obey the heat conduction inequality

(5) q ( u , T , h ) | h = 0 = 0 , ∂ h q ( u , T , h ) > 0 .

In [5], Dafermos and Hsiao considered the specific constitutive equations

(6) e = ∫ u ̄ u p ̃ ( ξ ) d ξ + c ( T − T ̄ ) , σ = p ̃ ( u ) + μ ( T − T ̄ ) , q = κ T x ,

where c > 0, κ > 0, μ ≠ 0, u ̄ and T ̄ are constants, and p ̃ ( u ) is a smooth function. The relation (6) can be regarded as the approximation of the general constitutive equations around the state ( u ̄ , T ̄ ) . Combining (3) and (6), Dafermos and Hsiao [5] obtained

(7) u t − v x = 0 , v t − p ̃ u ( u ) u x = κ c θ x , θ t − κ c θ x x = μ c θ u t ,

where θ ( t , x ) = c μ [ T ( t , x ) − T ̄ ] / κ . One differentiates the first two equations in (7) and scales the independent variable x in the resulting to get

(8) u t t − ( p u ( u ) u x ) x = θ x x , θ t − θ x x = f ( θ ) u t ,

with p u ( u ) = c p ̃ u ( u ) / κ and f ( θ ) = μ θ / c κ . Since the compatibility relation (4) holds only when T ̄ = 0 in (6), to fully comply with (4), Hrusa and Messaoudi [6] adopted the following expression of e in (6)

(9) e = ∫ u ̄ u p ̃ ( ξ ) d ξ + c ( T − T ̄ ) − μ T ̄ u .

For the expression of e in (9), one obtains (8) with f ( θ ) = μ ( κ θ + c μ T ̄ ) / ( κ c κ ) . Taking the common polytropic relation p(u) = u ^1+2a/(1 + 2a) with a > 0 in (8) yields the system (1). In addition, a general form of system (8) was introduced by Slemrod [7]

(10) u t t − ( a ( u , θ ) u x ) x + ( b ( u , θ ) θ x ) x = 0 , c ( u , θ ) θ t − d ( u , θ ) θ x x + b ( u , θ ) f ( θ ) u t = 0 .

Under the assumption that p _u(u) > 0 for all u under consideration, the first equation of (8) is the strictly hyperbolic type. Based on the strict hyperbolicity, systems (8) and (10) have been widely investigated, see works on the singularity formations of smooth solutions [5], [6], [8], on the local existence and uniqueness of smooth solutions to the Cauchy problem [7], [9], [10] and the initial-boundary value problems [11], [12], [13], [14], [15], [16], on the global existence and asymptotic stability of smooth solutions to the Cauchy problem [17], [18], [19] and the initial-boundary value problems [7], [20], [21], [22], [23]. Moreover, the reader may consult [8], [24], [25], [26] for the discussions to the multidimensional equations and [27], [28], [29], [30], [31] for the works in nonlinear thermoelasticity with second sound. For more relevant results about the thermoelasticity and the related hyperbolic-parabolic coupled systems, one can refer to [32], [33], [34], [35], [36], [37], [38] and references therein.

If the function p _u(u) allows to disappear or change sign in the region of u under consideration, the first equation of (8) may degenerate and the corresponding problems are very different and complicated. To the best of our knowledge, the research results involving degenerate cases are quite limited. The main difficulty is that the system is not continuously differentiable due to the occurrence of degeneracy and then the classical method for dealing with the strictly hyperbolic-parabolic coupled systems is invalid. Some existence results for the initial value problems of the degenerate hyperbolic systems can be found in [39], [40], [41], [42], [43]. In [42], [44], [45], the existence and uniqueness for some quasilinear wave equations including (1) with θ = 0 is studied. In particular, the solution is constructed in C 0 ∞ class by using Nash–Moser theorem. It is noted that if θ = 0, the support of solutions to (1) is included in the support of initial values. For thermoelastic case (θ ≠ 0), the support of solution is not compact, even if that of initial data is compact. This fact also motivates to study (1) with initial data degenerating at spatial infinity. On the other hand, corresponding to the above discussed degenerate hyperbolic problem, the research on the degenerate parabolic problem of (10) with a > 0 while d ≥ 0 are also rare. A related work can be referred to [46], in which the authors studied a linear system of thermoelasticity with a degenerated second order operator in the parabolic equation.

In the present paper, we consider the local existence and uniqueness of classical solutions to the Cauchy problem of the hyperbolic-parabolic coupled system (1) with (2). If the initial data u ₀(x) in (2) satisfies

(11) u 0 ( x ) ≥ c 0 > 0 ,

for some constant c ₀, the local solvability for problem (1), (2) can be obtained by the classical method of hyperbolic-parabolic coupled systems since the first equation of (1) is strictly hyperbolic near t = 0. In this paper, we relax the condition (11) to allow the initial data u ₀(x) and then the first equation of (1) to degenerate at spacial infinity. We comment that, although the degradation only occurs at infinity, we can not solve this problem by the previous theory for the strictly hyperbolic-parabolic coupled system, because the coefficients of the corresponding first-order hyperbolic system are not uniformly bounded. We verify that the existence and uniqueness of classical solutions for small enough time by the contraction mapping principle via suitable weighted L ^∞ estimates. The idea used here is inspired by a recent work of the second author [47] for studying the Cauchy problem to a quasilinear wave equation with far field degeneracy. Compared with a single hyperbolic equation, the key of this paper is to reasonably deal with the coupling of hyperbolicity and parabolicity, which makes the current problem more complicated.

The rest of the paper is organized as follows. Section 2 is devoted to providing the assumptions on the initial data and then stating the main result. The choice of the constants of the paper is also presented in this section. In Section 3, we analyze some properties of the characteristic curves for the hyperbolic system and of the fundamental solution for the heat equation. The proof of the main theorem is given in Section 4. In Subsection 4.1, we describe the admissible function class with weighted L ^∞-norm and then construct an integration iteration mapping for the Cauchy problem. In Subsection 4.2, we show that the iteration mapping maps the admissible function set into itself and is a contraction under the weighted L ^∞-norm. In Subsection 4.3, we verify that the limit vector function also belongs to the function set and then complete the proof of the theorem.

2 The assumptions and main result

Let α and β be two constants satisfying

(12) 0 < α ≤ β , 0 < 2 + a 3 α ≤ β .

We note that the first inequality in (12) is not necessary if the number a ≥ 1, while the second one is not necessary if a ≤ 1. Denote

(13) γ = 0 , a ≥ 1 , ( 1 − a ) α , 0 < a < 1 .

We assume that the initial data (u ₀(x), u ₁(x), θ ₀(x)) satisfy

(14) u 0 ( x ) ∈ C 2 ( R ) , u 1 ( x ) ∈ C 1 ( R ) , θ 0 ( x ) ∈ C 2 ( R ) ,

and

(15) ( A 1 ) : 2 K 0 ⟨ x ⟩ − α ≤ u 0 ( x ) ≤ K 1,0 , ( A 2 ) : u 0 a | u 0 ′ | ≤ K 2,0 ⟨ x ⟩ − β , ( A 3 ) : | u 1 | ≤ K 3,0 ⟨ x ⟩ − β , ( A 4 ) : d d x u 1 ± u 0 a u 0 ′ − θ 0 ≤ K 5,0 ⟨ x ⟩ − γ , ( A 5 ) : | θ 0 ( x ) | ≤ M 1,0 ⟨ x ⟩ − β , ( A 6 ) : | θ 0 ′ | ≤ M 2,0 ⟨ x ⟩ − β , ( A 7 ) : | θ 0 ″ | ≤ M 3,0 ⟨ x ⟩ − γ ,

where ⟨ x ⟩ = 1 + x 2 , and K ₀, K _i,0, M _i,0 are positive constants. It follows from the condition (A ₁) in (15) that the first equation in (1) is degenerate at spatial infinity. The conditions (A ₁) and (A ₂) imply that

(16) ⟨ x ⟩ β − a α | u 0 ′ | ≤ K 2,0 ( 2 K 0 ) a .

Since the number (β − aα) may be less than zero, the function u 0 ′ is allowed to be unbounded when x goes to infinity. Furthermore, the conditions (A ₂), (A ₃) and (A ₅) mean that

(17) | u 1 ± u 0 a u 0 ′ − θ 0 | ≤ K 4,0 ⟨ x ⟩ − β ,

where K _4,0 = K _2,0 + K _3,0 + M _1,0.

The main conclusion of this paper is the following.

Theorem 1.

Let f be a given C ¹ function satisfying | f ( z ) | + | f ′ ( z ) | ≤ f ̄ for a constant f ̄ in a finite region of z. Suppose that the initial conditions (14) and (15) hold. Then there exists a small number T > 0 such that the Cauchy problem (1), (2) admits a unique classical solution (u, θ)(t, x) on [ 0 , T ] × R . Moreover, the solution (u, θ)(t, x) satisfies

(18) ( B 1 ) : K 0 ⟨ x ⟩ − α ≤ u ( t , x ) ≤ K 1 , ( B 2 ) : u a | u x | ≤ K 2 ⟨ x ⟩ − β , o r ⟨ x ⟩ β − a α | u x | ≤ K 2 K 0 a , ( B 3 ) : | u t | ≤ K 3 ⟨ x ⟩ − β , ( B 4 ) : | u t ± u a u x − θ | ≤ K 4 ⟨ x ⟩ − β , ( B 5 ) : | ∂ x u t ± u a u x − θ | ≤ K 5 ⟨ x ⟩ − γ , ( B 6 ) : | ∂ t u t ± u a u x − θ | ≤ K 6 ⟨ x ⟩ − γ , ( B 7 ) : | θ ( t , x ) | ≤ M 1 ⟨ x ⟩ − β , ( B 8 ) : | θ x | ≤ M 2 ⟨ x ⟩ − β , ( B 9 ) : | θ x x | ≤ M 3 ⟨ x ⟩ − γ , ( B 10 ) : | θ t | ≤ M 4 ⟨ x ⟩ − γ ,

where K _i, M _i are positive constants depending only on the constants a , α , β , f ̄ , K 0 , K i , 0 , M i , 0 .

Remark 1.

It is obvious by (B ₂) in (18) that, if β < aα, u _x is not uniformly bounded and then u is only locally Lipschitz continuous with respect to x, while the combination u ^a u _x is uniformly bounded. The example of u ₀ satisfying the decay conditions in (A ₁), (A ₂) and (A ₄) with a > 1 and whose derivative is not bounded is

u 0 ( x ) = 〈 x 〉 − α ( 2 + sin 〈 x 〉 λ + 1 ) ,

where α and λ fulfil the condition α < λ ≤ (1 + a)α/2. One can easily compute that

〈 x 〉 − α ≤ u 0 ( x ) ≤ 3 〈 x 〉 − α , | u 0 a u 0 ′ ( x ) | ≤ C 1 〈 x 〉 − α ( 1 + a ) + λ ≤ C 1 〈 x 〉 − α ( 1 + a ) 2 ≤ C 1 〈 x 〉 − α ( 2 + a ) 3 , d d x ( u 0 a u 0 ′ ( x ) ) ≤ C 2 〈 x 〉 − α ( a + 1 ) + 2 λ ≤ C 2 ,

and

lim n → ∞ | u 0 ′ ( n π ) | = ∞ ,

where C ₁ and C ₂ are positive constants.

Remark 2.

In [47], the second author construct a local solution to the equation

u t t − u 2 a u x x − F ( u ) u x = 0 ,

under the assumption that F decays faster that u ^a at spacial infinity, which is called Levi condition in the study of the weakly hyperbolic equation. In stead of (12), to control F(u)u _x, the following slightly stronger condition is assumed.

0 < α ≤ β , 0 < 2 + a 2 α ≤ β .

We shall use the contraction mapping principle to show Theorem 1. The key is to establish the weighted estimates in (18). We utilize the method of characteristic to derive the estimates for u and employ the fundamental solution of one-dimensional heat equation to acquire the estimates for θ. It is emphasized that the purpose of requiring the weighted estimates in (18) is to obtain a consistent T. If we only have the usual estimates, the constants K _i and M _i will depend on the spatial variable x, and then T may go to zero when x tends to infinity. This approach can also be applied to the more general hyperbolic-parabolic coupled systems under some suitable conditions, for example system (10).

For the convenience of readers, we here first give the results of our choices of the constants K _i, M _i and T, which come from the construction process in Sections 3 and 4.

(19) K 1 = 2 K 1,0 , K 2 = 2 1 + a K 2,0 + 3 K 4,0 + ( 3 + a ) 2 a + β K 4,0 , K 3 = 2 1 + β ( K 4,0 + 4 M 1,0 ) , K 4 = 2 1 + β K 4,0 , K 5 = 2 2 + γ K 5,0 , K 6 = 2 2 + a + β K 1,0 a K 5,0 + a 2 2 + 2 β K 4,0 ( K 4,0 + 4 M 1,0 ) K 0 + 2 1 + β f ̄ ( K 4,0 + 4 M 1,0 ) + 2 3 + a + β K 1,0 a M 2,0 ,

and

(20) M 1 = 2 3 + β M 1,0 , M 2 = 2 3 + β M 2,0 , M 3 = 2 1 + γ ( 3 + P γ ) M 3,0 , M 4 = 2 1 + γ ( 3 + P γ ) M 3,0 + f ̄ 2 1 + β ( K 4,0 + 4 M 1,0 ) .

Here and below we use P _κ to denote a positive constant such that there holds

(21) ξ κ 2 ⁡ exp − ξ 4 ≤ P κ ⁡ exp − ξ 16 ,

for ξ ≥ 0 and κ ≥ 0. The inequality (21) arises from the estimates for the fundamental solution of the heat equation. Now we choose the number T ≤ 1 to be sufficiently small such that the following inequalities hold simultaneously

(22) P β T β 2 ≤ 1 , P 1 + β T β 2 ≤ 1 , a K 2 K 0 T ≤ 1 2 , K 3 K 0 T ≤ 1 2 , K 3 T ≤ K 1,0 , K 1 a T ≤ 1 2 , a ( K 4 + M 1 ) K 4 K 0 + f ̄ ( K 4 + M 1 ) + K 1 a M 2 T ≤ K 4,0 , f ̄ ( K 4 + M 1 ) T ≤ M 1,0 , 2 f ̄ ( K 4 + M 1 ) T ≤ M 2,0 , f ̄ M 2 ( 1 + K 4 + M 1 ) + K 5 ( 1 + P 1 + γ ) T ≤ M 3,0 , a ( 2 K 4 K 5 + K 4 M 2 + K 5 M 1 + K 2 M 2 ) K 0 + a K 2 K 4 ( K 4 + M 1 ) K 0 2 + a T ≤ 1 2 K 5,0 ,

and

(23) f ̄ M 2 ( K 4 + M 1 ) + f ̄ ( K 5 + M 2 ) + K 1 a M 3 T ≤ 1 2 K 5,0 , a K 3 K 0 + a K 3 K 4 K 0 2 + 2 f ̄ ( K 3 + 1 ) + K 1 a + a M 2 K 1 a − 1 + a K 1 a − 1 K 5 T ≤ 1 12 , f ̄ ( K 3 + 1 ) T ≤ 1 16 .

We comment that, according to the choices of constants K _i, M _i and P _κ in (19)–(21), the number T is indeed a consistent constant, independent of the spatial variable x.

3 Preliminaries for the hyperbolic and parabolic equations

In this section, we provide some basic knowledge about the wave and heat equations, including the estimates of characteristic curves and the fundamental solution of heat equation.

3.1 Basis of hyperbolic equation

Putting the second equation in (1) into the first equation, we achieve an equivalent system for smooth solutions

(24) u t t − u 2 a u x x − 2 a u 2 a − 1 u x 2 = θ t − f ( θ ) u t , θ t − θ x x = f ( θ ) u t .

In order to deal with the term θ _t in the first equation of (24), we introduce

(25) R = u t + u a u x − θ , S = u t − u a u x − θ ,

so that

(26) u t = R + S 2 + θ , u x = R − S 2 u a .

In terms of variables (R, S, u, θ), system (24) can be expressed as

(27) R t − u a R x = a ( R + θ ) ( R − S ) 2 u − 1 2 f ( θ ) ( R + S + 2 θ ) + u a θ x , S t + u a S x = a ( S + θ ) ( S − R ) 2 u − 1 2 f ( θ ) ( R + S + 2 θ ) − u a θ x , u t = R + S 2 + θ , θ t − θ x x = f ( θ ) R + S 2 + θ .

Corresponding to (2), the initial values of (R, S, u, θ) are

(28) ( R ( t , x ) , S ( t , x ) , u ( t , x ) , θ ( t , x ) ) | t = 0 = ( R 0 ( x ) , S 0 ( x ) , u 0 ( x ) , θ 0 ( x ) ) ,

where

(29) R 0 ( x ) = u 1 ( x ) + u 0 a u 0 ′ ( x ) − θ 0 ( x ) , S 0 ( x ) = u 1 ( x ) − u 0 a u 0 ′ ( x ) − θ 0 ( x ) .

In view of (15) and (17), it suggests that

(30) | R 0 ( x ) | , | S 0 ( x ) | ≤ K 4,0 〈 x 〉 − β , | R 0 ′ ( x ) | , | S 0 ′ ( x ) | ≤ K 5,0 〈 x 〉 − γ .

Clearly, the two eigenvalues of the first two equations in system (27) are λ _± = ±u ^a and the corresponding characteristics x _±(t) = x _±(t; ξ, η) passing through a point (ξ, η) are

(31) x ± ( t ) = η ± ∫ ξ t u a ( s , x ± ( s ) ) d s .

Along the characteristics x _±(t), the equations for R and S in (27) can be rewritten as

(32) d d t R ( t , x − ( t ) ) = a ( R + θ ) ( R − S ) 2 u − 1 2 f ( θ ) ( R + S + 2 θ ) + u a θ x ( t , x − ( t ) ) , d d t S ( t , x + ( t ) ) = a ( S + θ ) ( S − R ) 2 u − 1 2 f ( θ ) ( R + S + 2 θ ) − u a θ x ( t , x + ( t ) ) .

For the characteristics x _±(t), we have

Lemma 3.1.

Assume that u(t, x) is a C 1 ( [ 0 , T ] × R ) function satisfying

(33) K 0 〈 x 〉 − α ≤ u ( t , x ) ≤ K 1 , 0 ≤ u a | u x | ≤ K 2 〈 x 〉 − β ,

where K _i are constants given in (19) and T satisfies inequalities in (22). Then there hold

(34) 2 − κ 2 ⟨ η ⟩ κ ≤ ⟨ x ± ( t ) ⟩ κ ≤ 2 κ 2 ⟨ η ⟩ κ ,

for κ ≥ 0 and

(35) x ± ( t 1 ; ξ 1 , η 1 ) − x ± ( t 2 ; ξ 2 , η 2 ) ≤ 2 1 + A 1 a | t 1 − t 2 | + | ξ 1 − ξ 2 | + | η 1 − η 2 | ,

for any t ₁, t ₂, ξ ₁, η ₂ ∈ [0, T] and η 1 , η 2 ∈ R . This means that x _±(t; ξ, η) are uniformly Lipschitz continuous with respect to t, ξ, η. Furthermore, the characteristics x _±(t; ξ, η) are differentiable with respect to η and

(36) ∂ x ± ( t ; ξ , η ) ∂ η ≤ 2 .

Proof.

We first show (34). It infers from (31) and (33) that

(37) | x ± ( t ) | 2 ≤ 2 η 2 + 2 K 1 a T 2 .

On the other hand, one has ( x ± ( t ) − η ) 2 ≤ K 1 a T 2 , as a consequence

(38) | x ± ( t ) | 2 + η 2 − K 1 a T 2 ≤ 2 x ± ( t ) η ≤ 2 | x ± ( t ) | 2 + η 2 2 .

Combining (37) and (38) yields η 2 2 − K 1 a T 2 ≤ | x ± ( t ) | 2 ≤ 2 η 2 + 2 K 1 a T 2 , which implies by K 1 a T ≤ 1 2 that

1 + η 2 2 ≤ 1 + | x ± ( t ) | 2 ≤ 2 ( η 2 + 1 ) .

Hence we get the desired inequalities in (34).

We next show (35). For the case t ₁ = t ₂ = t, it derives by (31) and (33) that

(39) x ± ( t ; ξ 1 , η 1 ) − x ± ( t ; ξ 2 , η 2 ) ≤ | η 1 − η 2 | + ∫ ξ 1 ξ 2 u a ( s , x ± ( s ; ξ 1 , η 1 ) ) d s + ∫ 0 T u a ( s , x ± ( s ; ξ 1 , η 1 ) ) − u a ( s , x ± ( s ; ξ 2 , η 2 ) ) d s ≤ | η 1 − η 2 | + K 1 a | ξ 1 − ξ 2 | + ∫ 0 T a 〈 x ± ( s ) 〉 β u a u x 〈 x ± ( s ) 〉 α u L ∞ ⋅ x ± ( s ; ξ 1 , η 1 ) − x ± ( s ; ξ 2 , η 2 ) d s ≤ | η 1 − η 2 | + K 1 a | ξ 1 − ξ 2 | + a K 2 K 0 ∫ 0 T x ± ( s ; ξ 1 , η 1 ) − x ± ( s ; ξ 2 , η 2 ) d s ,

from which and the Gronwall inequality one achieves

(40) x ± ( t ; ξ 1 , η 1 ) − x ± ( t ; ξ 2 , η 2 ) ≤ 2 1 + K 1 a | ξ 1 − ξ 2 | + | η 1 − η 2 | ,

by the fact a K 2 K 0 T ≤ 1 2 . For the case t ₁ ≠ t ₂, we have by (31) and (40)

(41) x ± ( t 1 ; ξ 1 , η 1 ) − x ± ( t 2 ; ξ 2 , η 2 ) ≤ x ± ( t 1 ; ξ 1 , η 1 ) − x ± ( t 2 ; ξ 1 , η 1 ) + x ± ( t 2 ; ξ 1 , η 1 ) − x ± ( t 2 ; ξ 2 , η 2 ) ≤ 2 1 + K 1 a | t 1 − t 2 | + | ξ 1 − ξ 2 | + | η 1 − η 2 | .

Finally, we prove (36). The differentiability of x _±(t; ξ, η) with respect to η is a classical known result. Differentiating (31) with respect to η gives

(42) ∂ x ± ( t ; ξ , η ) ∂ η ≤ 1 + a K 2 K 0 ∫ 0 T ∂ x ± ( s ; ξ , η ) ∂ η d s ,

which together with the Gronwall inequality and the fact a K 2 K 0 T ≤ 1 2 leads to (36).□

3.2 Basis of heat equation

In this subsection, we introduce some known results for the heat equation. Let b(t, x) ∈ C ¹ be a given function on [ 0 , T ] × R . We consider the following initial value problem for the heat equation

(43) θ t − θ x x = b ( t , x ) , θ ( t , x ) | t = 0 = θ 0 ( x ) .

The fundamental solution of the heat equation is

(44) H ( t , x ) = 1 4 π t exp − x 2 4 t .

Then the solution of (43) can be expressed as

(45) θ ( t , x ) = ∫ − ∞ + ∞ H ( t , z ) θ 0 ( x − z ) d z + ∫ 0 t ∫ − ∞ + ∞ H ( t − τ , z ) b ( τ , x − z ) d z d τ .

Furthermore, we have

(46) θ x ( t , x ) = ∫ − ∞ + ∞ H ( t , z ) θ 0 ′ ( x − z ) d z + ∫ 0 t ∫ − ∞ + ∞ ∂ H ( t − τ , z ) ∂ z b ( τ , x − z ) d z d τ ,

(47) θ x x ( t , x ) = ∫ − ∞ + ∞ H ( t , z ) θ 0 ″ ( x − z ) d z + ∫ 0 t ∫ − ∞ + ∞ ∂ H ( t − τ , z ) ∂ z b x ( τ , x − z ) d z d τ .

Set

(48) H σ ( t , x ) = 1 4 π t − σ 2 ⁡ exp − x 2 16 t .

It is not difficult to see that

(49) H ( t − ς , z ) ≤ H 1 ( t − ς , z ) , ∂ z H ( t − ς , z ) ≤ H 2 ( t − ς , z ) .

Moreover, there holds

(50) ∫ − ∞ + ∞ H σ ( t , z ) d z = 2 t 1 − σ 2 , ∫ 0 t ∫ − ∞ + ∞ H σ ( t − τ , z ) d z d τ = 4 3 − σ t 3 − σ 2 ( σ < 3 ) .

In addition, we calculate

(51) 〈 x 〉 κ 〈 x − z 〉 κ = 1 + x 2 1 + ( x − z ) 2 κ 2 ≤ 1 + 2 ( x − z ) 2 + 2 z 2 1 + ( x − z ) 2 κ 2 ≤ 2 κ ( 1 + | z | κ ) ,

for κ ≥ 0. Recalling (21), (44) and (48) arrive at

(52) | z | κ H ( t , z ) = z 2 t κ 2 ⋅ 1 4 π t exp − z 2 4 t ⋅ t κ 2 ≤ t κ 2 P κ H 1 ( t , z ) ,

(53) | z | κ ∂ H ( t , z ) ∂ z = t 1 + κ 2 2 t 4 π t z 2 t 1 + κ 2 ⁡ exp − z 2 4 t ≤ t κ 2 P 1 + κ H 2 ( t , z ) .

4 Proof of the main theorem

In this section, we shall show the main theorem of the paper by utilizing the contraction mapping principle.

4.1 The iterative mapping

We use the notation Σ(T) to represent the function class which incorporates all vector functions F = ( f 1 , f 2 , f 3 , f 4 ) : [ 0 , T ] × R → R 4 satisfying the following properties:

(54) ( P 1 ) : f i ( t , x ) ( i = 1 , ⋅ , 4 ) a r e C 1 f u n c t i o n s ; ( P 2 ) : f 4 x is continuously differentiable with respect to x ; ( P 3 ) : ( f 1 , f 2 , f 3 , f 4 ) ( 0 , x ) = ( u 0 ( x ) , R 0 ( x ) , S 0 ( x ) , θ 0 ( x ) ) , ∀ x ∈ R ; ( P 4 ) : K 0 〈 x 〉 − α ≤ f 1 ( t , x ) ≤ K 1 , f 1 a | f 1 x | ≤ K 2 〈 x 〉 − β , | f 1 t | ≤ K 3 〈 x 〉 − β , | f 2 ( t , x ) | , | f 3 ( t , x ) | ≤ K 4 〈 x 〉 − β , | f 2 x | , | f 3 x | ≤ K 5 〈 x 〉 − γ , | f 2 t | , | f 3 t | ≤ K 6 〈 x 〉 − γ , | f 4 ( t , x ) | ≤ M 1 〈 x 〉 − β , | f 4 x | ≤ M 2 〈 x 〉 − β , | f 4 xx | ≤ M 3 〈 x 〉 − γ , | f 4 t | ≤ M 4 〈 x 〉 − γ ,

where the constants K _i and M _i are given in (19) and (20), and the positive number T is sufficiently small that all inequalities in (22) hold.

Let vector function ( u ̃ , R ̃ , S ̃ , θ ̃ ) be any element in Σ(T). We consider the following Cauchy problem of the linear hyperbolic-parabolic coupled system

(55) R t − u ̃ a R x = a ( R ̃ + θ ̃ ) ( R ̃ − S ̃ ) 2 u ̃ − 1 2 f ( θ ̃ ) ( R ̃ + S ̃ + 2 θ ̃ ) + u ̃ a θ ̃ x , S t + u ̃ a S x = a ( S ̃ + θ ̃ ) ( S ̃ − R ̃ ) 2 u ̃ − 1 2 f ( θ ̃ ) ( R ̃ + S ̃ + 2 θ ̃ ) − u ̃ a θ ̃ x , θ t − θ x x = f ( θ ̃ ) R ̃ + S ̃ 2 + θ ̃ ,

supplemented with

(56) R ( 0 , x ) = R 0 ( x ) , S ( 0 , x ) = S 0 ( x ) , θ ( 0 , x ) = θ 0 ( x ) .

For any point ( ξ , η ) ∈ [ 0 , T ] × R , integrating the first two equations in (55) from 0 to ξ and applying (45), we achieve

(57) R ( ξ , η ) = R 0 ( x ̃ − ( 0 ) ) + ∫ 0 ξ a ( R ̃ + θ ̃ ) ( R ̃ − S ̃ ) 2 u ̃ − 1 2 f ( θ ̃ ) ( R ̃ + S ̃ + 2 θ ̃ ) + u ̃ a θ ̃ x ( t , x ̃ − ( t ) ) d t ,

(58) S ( ξ , η ) = S 0 ( x ̃ + ( 0 ) ) + ∫ 0 ξ a ( S ̃ + θ ̃ ) ( S ̃ − R ̃ ) 2 u ̃ − 1 2 f ( θ ̃ ) ( R ̃ + S ̃ + 2 θ ̃ ) − u ̃ a θ ̃ x ( t , x ̃ + ( t ) ) d t ,

(59) θ ( ξ , η ) = ∫ − ∞ + ∞ H ( ξ , z ) θ 0 ( η − z ) d z + ∫ 0 ξ ∫ − ∞ + ∞ H ( ξ − τ , z ) f ( θ ̃ ) R ̃ + S ̃ 2 + θ ̃ ( τ , η − z ) d z d τ .

Here the curves x ̃ ± ( t ) = x ̃ ± ( t ; ξ , η ) in (57) and (58) are defined as

(60) d d t x ̃ ± ( t ; ξ , η ) = ± u ̃ a ( t , x ̃ ± ( t ; ξ , η ) ) , x ̃ ± ( ξ ; ξ , η ) = η .

After obtaining the functions (R, S, θ)(t, x), one can define the function u(t, x) by

(61) u ( ξ , η ) = u 0 ( η ) + ∫ 0 ξ R + S 2 + θ ( τ , η ) d τ .

We note that (57)–(61) determine an iteration mapping T :

(62) ( u , R , S , θ ) = T ( u ̃ , R ̃ , S ̃ , θ ̃ ) .

We shall show that (u, R, S, θ) ∈ Σ(T), the mapping T is a contraction under the norm of L ^∞, and the fixed point of the mapping T also belongs to Σ(T).

4.2 Properties of iteration mapping

Lemma 4.1.

Let ( u ̃ , R ̃ , S ̃ , θ ̃ ) ( t , x ) be any element in Σ(T). Then the vector function (u, R, S, θ)(t, x) determined by (57)–(61) belongs to Σ(T). This means that T maps Σ(T) to Σ(T).

Proof.

The proof of the lemma is divided into five steps.

Step 1. The estimates of R and S. For any point ( ξ , η ) ∈ [ 0 , T ] × R , we employ (57), (34) with κ = β, (30) and (54) to acquire

(63) | 〈 η 〉 β R ( ξ , η ) | ≤ | 〈 η 〉 β R 0 ( x ̃ − ( 0 ) ) | + ∫ 0 ξ 〈 η 〉 β a ( R ̃ + θ ̃ ) ( R ̃ − S ̃ ) 2 u ̃ − 1 2 f ( θ ̃ ) ( R ̃ + S ̃ + 2 θ ̃ ) + u ̃ a θ ̃ x ( t , x ̃ − ( t ) ) d t ≤ 2 β 2 | 〈 x ̃ − ( 0 ) 〉 β R 0 ( x ̃ − ( 0 ) ) | + 2 β 2 ∫ 0 ξ a 〈 x ̃ − ( t ) 〉 2 β | R ̃ + θ ̃ | ⋅ | R ̃ − S ̃ | 2 〈 x ̃ − ( t ) 〉 α u ̃ + 1 2 f ̄ 〈 x ̃ − ( t ) 〉 β | R ̃ + S ̃ + 2 θ ̃ | + u ̃ a 〈 x ̃ − ( t ) 〉 β | θ ̃ x | d t ≤ 2 β 2 K 4,0 + 2 β 2 ∫ 0 T a ( K 4 + M 1 ) ⋅ 2 K 4 2 K 0 + 1 2 f ̄ ( 2 K 4 + 2 M 1 ) + K 1 a M 2 d t ≤ 2 β 2 K 4,0 + 2 β 2 K 4,0 ≤ 2 1 + β K 4,0 = K 4 ,

due to the choice of T in (22). The above estimate is also valid for the function S.

Step 2. The estimates of θ and θ _x. By (59), (51) with κ = β and (54), we calculate

(64) | ⟨ η ⟩ β θ ( ξ , η ) | ≤ ∫ − ∞ + ∞ H ( ξ , z ) ⟨ η ⟩ β | θ 0 ( η − z ) | d z + ∫ 0 ξ ∫ − ∞ + ∞ H ( ξ − τ , z ) ⟨ η ⟩ β | f ( θ ̃ ) | ⋅ R ̃ + S ̃ 2 + θ ̃ ( τ , η − z ) d z d τ ≤ ∫ − ∞ + ∞ H ( ξ , z ) ⋅ 2 β ( 1 + | z | β ) ⟨ η − z ⟩ β | θ 0 ( η − z ) | d z + ∫ 0 ξ ∫ − ∞ + ∞ H ( ξ − τ , z ) ⋅ 2 β ( 1 + | z | β ) ⟨ η − z ⟩ β | f ( θ ̃ ) | ⋅ R ̃ + S ̃ 2 + θ ̃ ( τ , η − z ) d z d τ ≤ 2 β M 1,0 ∫ − ∞ + ∞ H ( ξ , z ) ( 1 + | z | β ) d z + 2 β f ̄ ( K 4 + M 1 ) ∫ 0 ξ ∫ − ∞ + ∞ H ( ξ − τ , z ) ( 1 + | z | β ) d z d τ ,

from which and (52), (50), one has

(65) | ⟨ η ⟩ β θ ( ξ , η ) | ≤ 2 β M 1,0 ∫ − ∞ + ∞ H 1 ( ξ , z ) ( 1 + T β 2 P β ) d z + 2 β f ̄ ( K 4 + M 1 ) ∫ 0 ξ ∫ − ∞ + ∞ H 1 ( ξ − τ , z ) ( 1 + T β 2 P β ) d z d τ ≤ 2 1 + β M 1,0 ∫ − ∞ + ∞ H 1 ( ξ , z ) d z + 2 1 + β f ̄ ( K 4 + M 1 ) ∫ 0 ξ ∫ − ∞ + ∞ H 1 ( ξ − τ , z ) d z d τ = 2 1 + β M 1,0 ⋅ 2 + 2 1 + β f ̄ ( K 4 + M 1 ) ⋅ 2 ξ ≤ 2 2 + β M 1,0 + 2 2 + β f ̄ ( K 4 + M 1 ) T ≤ 2 2 + β M 1,0 + 2 2 + β M 1,0 = M 1 .

Here we used the facts T β 2 P β ≤ 1 and f ̄ ( K 4 + M 1 ) T ≤ M 1,0 by (22).

To estimate the function θ _x, we combine (46) and (59) to get

(66) | ⟨ η ⟩ β θ η ( ξ , η ) | ≤ ∫ − ∞ + ∞ H ( ξ , z ) ⟨ η ⟩ β | θ 0 ′ ( η − z ) | d z + ∫ 0 ξ ∫ − ∞ + ∞ ∂ H ( t − τ , z ) ∂ z ⟨ η ⟩ β | f ( θ ̃ ) | ⋅ R ̃ + S ̃ 2 + θ ̃ ( τ , η − z ) d z d τ ≤ 2 β ∫ − ∞ + ∞ H ( ξ , z ) ( 1 + | z | β ) ⟨ η − z ⟩ β | θ 0 ′ ( η − z ) | d z + 2 β f ̄ ∫ 0 ξ ∫ − ∞ + ∞ ∂ H ( t − τ , z ) ∂ z ( 1 + | z | β ) ⟨ η − z ⟩ β R ̃ + S ̃ 2 + θ ̃ ( τ , η − z ) d z d τ ≤ 2 β M 2,0 ∫ − ∞ + ∞ H ( ξ , z ) ( 1 + | z | β ) d z + 2 β f ̄ ( K 4 + M 1 ) ∫ 0 ξ ∫ − ∞ + ∞ ∂ H ( t − τ , z ) ∂ z ( 1 + | z | β ) d z d τ .

We utilize (49) and (50) and (52) and (53) again to gain

(67) | ⟨ η ⟩ β θ η ( ξ , η ) | ≤ 2 β M 2,0 ∫ − ∞ + ∞ H 1 ( ξ , z ) ( 1 + T β 2 P β ) d z + 2 β f ̄ ( K 4 + M 1 ) ∫ 0 ξ ∫ − ∞ + ∞ H 2 ( t − τ , z ) ( 1 + T β 2 P 1 + β ) d z d τ ≤ 2 1 + β M 2,0 ∫ − ∞ + ∞ H 1 ( ξ , z ) d z + 2 1 + β f ̄ ( K 4 + M 1 ) ∫ 0 ξ ∫ − ∞ + ∞ H 2 ( t − τ , z ) d z d τ ≤ 2 2 + β M 2,0 + 2 2 + β ⋅ 2 f ̄ ( K 4 + M 1 ) T ≤ 2 2 + β M 2,0 + 2 2 + β M 2,0 = M 2 .

Here we employed the facts T β 2 P 1 + β ≤ 1 and 2 f ̄ ( K 4 + M 1 ) T ≤ M 2,0 by (22).

Step 3. The estimates of u, u _t, u ^a u _x and u _x. We recall the definition of u in (61) to deduce by (63) and (65)

(68) u ( ξ , η ) ≤ | u 0 ( η ) | + ∫ 0 ξ | R | + | S | 2 + | θ | d τ ≤ K 1,0 + ( K 4 + M 1 ) T ≤ K 1 .

On the other hand, one finds that

(69) ⟨ η ⟩ α u ( ξ , η ) ≥ ⟨ η ⟩ α u 0 ( η ) − ∫ 0 ξ ⟨ η ⟩ β | R | + ⟨ η ⟩ β | S | 2 + ⟨ η ⟩ β | θ | d τ ≥ 2 K 0 − ( K 4 + M 1 ) T = 2 K 0 − K 3 T ≥ 2 K 0 − K 0 = K 0 .

Combining (68) and (69) gives

(70) K 0 ⟨ η ⟩ − α ≤ u ( ξ , η ) ≤ K 1 .

We differentiate (61) with respect to ξ to achieve u ξ ( ξ , η ) = R + S 2 + θ , from which and (63), (65), (19) and (20) one gets

(71) | u ξ ( ξ , η ) | ≤ 〈 η 〉 β | R | + 〈 η 〉 β | S | 2 + 〈 η 〉 β | θ | 〈 η 〉 − β ≤ ( K 4 + M 1 ) 〈 η 〉 − β = K 3 〈 η 〉 − β .

To estimate the term u ^a u _x, we compute by (61)

(72) u a u η = u 0 ( η ) + ∫ 0 ξ R + S 2 + θ ( τ , η ) d τ a u 0 ′ ( η ) + u 0 ( η ) + ∫ 0 ξ R + S 2 + θ ( τ , η ) d τ a ∫ 0 ξ R η + S η 2 + θ η ( τ , η ) d τ ≕ I 1 + I 2 .

For the term I ₁, it infers from (63), (65) and (15) that

(73) | I 1 | ≤ 2 a u 0 a + ∫ 0 ξ | R | + | S | 2 + | θ | ( τ , η ) d τ a | u 0 ′ | ≤ 2 a u 0 a + [ ( K 4 + M 1 ) T ] a 〈 η 〉 − a β | u 0 ′ | = 2 a 1 + ( K 3 T ) a 1 〈 η 〉 β u 0 a u 0 a | u 0 ′ | ≤ 2 a 1 + K 3 T K 0 a ⋅ K 2,0 〈 η 〉 − β ≤ 2 1 + a K 2,0 〈 η 〉 − β .

For the second term in I ₂, we see by (55) that

(74) ∫ 0 ξ R η + S η 2 + θ η ( τ , η ) d τ ≤ ∫ 0 ξ R τ − S τ 2 u ̃ a d τ + ∫ 0 ξ a ( | R ̃ | + | S ̃ | + 2 | θ ̃ | ) | R ̃ − S ̃ | 4 u ̃ 1 + a d τ + ∫ 0 ξ ( | θ ̃ η | + | θ η | ) d τ ≕ I 2,1 + I 2,2 + I 2,3 .

The estimate of I _2,3 can be directly acquired by (54) and (67)

(75) I 2,3 ≤ ∫ 0 ξ 2 M 2 〈 η 〉 − β d τ ≤ 2 M 2 T 〈 η 〉 − β = 2 M 2 T K 1 a 〈 η 〉 − β u 0 a ≤ 2 K 4,0 〈 η 〉 − β u 0 a ,

by the fact M 2 K 1 a T ≤ K 4,0 . To estimate I _2,1 and I _2,2, we first derive the upper and lower bounds of u ̃ / u 0 . Recalling | u ̃ t ( t , x ) | ≤ K 3 ⟨ x ⟩ − β and u ̃ ( 0 , x ) = u 0 ( x ) , from which one obtains u 0 − K 3 T ⟨ x ⟩ − β ≤ u ̃ ≤ u 0 + K 3 T ⟨ x ⟩ − β . Thus by the fact 2K ₃ T ≤ K ₀

(76) u ̃ u 0 ≤ 1 + K 3 T 〈 x 〉 β u 0 ≤ 1 + K 3 T K 0 ≤ 2 , u ̃ u 0 ≥ 1 − K 3 T 〈 x 〉 β u 0 ≥ 1 − K 3 T K 0 ≥ 1 2 ,

which lead to 1 2 ≤ u ̃ u 0 ≤ 2 . From the above and (54), one can estimate the term I _2,2

(77) I 2,2 = a 4 ∫ 0 ξ ⟨ η ⟩ 2 β ( | R ̃ | + | S ̃ | + 2 | θ ̃ | ) | R ̃ − S ̃ | ⟨ η ⟩ β u ̃ ⋅ u 0 u ̃ a ⋅ ⟨ η ⟩ − β u 0 a d τ ≤ a 4 ∫ 0 ξ ( 2 K 4 + 2 M 1 ) ⋅ 2 K 4 K 0 ⋅ 2 a ⋅ ⟨ η ⟩ − β u 0 a d τ ≤ 2 a a K 3 K 4 K 0 T ⋅ ⟨ η ⟩ − β u 0 a .

For the term I _2,1, we apply the integration by parts and (54), (63) to deduce

(78) I 2,1 = R − S 2 u ̃ a − R 0 − S 0 2 u 0 a + ∫ 0 ξ a ( R − S ) u ̃ τ 2 u ̃ 1 + a d τ ≤ 2 K 4 〈 η 〉 − β 2 u 0 a ⋅ u 0 u ̃ a + 2 K 4,0 〈 η 〉 − β 2 u 0 a + ∫ 0 ξ a 〈 η 〉 β | R − S | ⋅ K 3 〈 η 〉 − β 2 〈 η 〉 β u ̃ ⋅ u 0 u ̃ a 1 u 0 a d τ ≤ 2 a K 4 + K 4,0 + 2 a K 4 K 3 K 0 T 〈 η 〉 − β u 0 a .

Putting (75) and (77) and (78) into (74) and employing the fact 2K ₃ T ≤ K ₀ yields

(79) ∫ 0 ξ R η + S η 2 + θ η ( τ , η ) d τ ≤ 3 K 4,0 + ( 3 + a ) 2 a + β K 4,0 ⟨ η ⟩ − β u 0 a .

Now we insert (73) and (79) into (44) to conclude

(80) | u a u η | ≤ 2 1 + a K 2,0 〈 η 〉 − β + 2 1 + a u 0 a ⋅ 3 K 4,0 + ( 3 + a ) 2 a + β K 4,0 〈 η 〉 − β u 0 a = 2 1 + a K 2,0 + 3 K 4,0 + ( 3 + a ) 2 a + β K 4,0 〈 η 〉 − β = K 2 〈 η 〉 − β .

The estimate of the function u _x can be achieved by (70) and (80)

K 0 a | u η | ≤ ( ⟨ η ⟩ α u ) a | u η | = ⟨ η ⟩ a α | u a u η | ≤ K 2 ⟨ η ⟩ a α − β ,

from which one has

(81) ⟨ η ⟩ β − a α | u η ( ξ , η ) | ≤ K 2 K 0 a .

Step 4. The estimates of R _x, S _x, R _t and S _t. Differentiating (57) with respect to η arrives at

(82) R η ( ξ , η ) = R 0 ′ ( x ̃ − ( 0 ) ) ∂ x ̃ − ( 0 ) ∂ η + ∫ 0 ξ I 3 ( t , x ̃ − ( t ) ) ∂ x ̃ − ( t ) ∂ η d t ,

where

(83) I 3 = a ( R ̃ x + θ ̃ x ) ( R ̃ − S ̃ ) + a ( R ̃ + θ ̃ ) ( R ̃ x − S ̃ x ) 2 u ̃ − a ( R ̃ + θ ̃ ) ( R ̃ − S ̃ ) 2 u ̃ 2 u ̃ x − 1 2 f ′ ( θ ̃ ) θ ̃ x ( R ̃ + S ̃ + 2 θ ̃ ) − 1 2 f ( θ ̃ ) ( R ̃ x + S ̃ x + 2 θ ̃ x ) + a u ̃ a − 1 u ̃ x θ ̃ x + u ̃ a θ ̃ x x .

We recall Lemma 3.1 and (30) to find that

(84) 〈 η 〉 γ | R η ( ξ , η ) | ≤ 2 〈 η 〉 γ | R 0 ′ ( x ̃ − ( 0 ) ) | + 2 ∫ 0 ξ 〈 η 〉 γ | I 3 ( t , x ̃ − ( t ) ) | d t ≤ 2 1 + γ K 5,0 + 2 1 + γ ∫ 0 ξ 〈 x ̃ − ( t ) 〉 γ | I 3 ( t , x ̃ − ( t ) ) | d t .

In addition, it follows by (83) that

(85) 〈 x ̃ − ( t ) 〉 γ | I 3 | ≤ 〈 x ̃ − ( t ) 〉 γ + α − 2 β ⋅ a 〈 x ̃ − ( t ) 〉 2 β [ | R ̃ x + θ ̃ x | ⋅ | R ̃ − S ̃ | + | R ̃ + θ ̃ | ⋅ | R ̃ x − S ̃ x | ] 2 〈 x ̃ − ( t ) 〉 α u ̃ + 〈 x ̃ − ( t ) 〉 γ + ( 2 + a ) α − 3 β ⋅ a 〈 x ̃ − ( t ) 〉 2 β | R ̃ + θ ̃ | ⋅ | R ̃ − S ̃ | 2 ( 〈 x ̃ − ( t ) 〉 α u ̃ ) 2 + a ( 〈 x ̃ − ( t ) 〉 β u a u ̃ x ) + f ̄ 2 〈 x ̃ − ( t ) 〉 γ − 2 β ⋅ 〈 x ̃ − ( t ) 〉 2 β | θ ̃ x | ( | R ̃ | + | S ̃ | + 2 | θ ̃ | ) + f ̄ 2 〈 x ̃ − ( t ) 〉 γ ( | R ̃ x | + | S ̃ x | + 2 | θ ̃ x | ) + a 〈 x ̃ − ( t ) 〉 γ + α − 2 β ⋅ 〈 x ̃ − ( t ) 〉 β u ̃ a u ̃ x 〈 x ̃ − ( t ) 〉 α u ̃ ( 〈 x ̃ − ( t ) 〉 β θ ̃ x ) + u ̃ a 〈 x ̃ − ( t ) 〉 γ | θ ̃ x x | ,

from which and (54), one has by the facts γ + α − 2β ≤ 0 and γ + (2 + a)α − 3β ≤ 0 in (12) and (13)

(86) ⟨ x ̃ − ( t ) ⟩ γ | I 3 | ≤ a ( 2 K 4 K 5 + M 2 K 4 + M 1 K 5 + K 2 M 2 ) K 0 + a ( K 4 + M 1 ) K 4 K 2 K 0 2 + a + f ̄ M 2 ( K 4 + M 1 ) + f ̄ ( K 5 + M 2 ) + K 1 a M 3 ≤ 1 T K 5,0 .

Here we used the last two inequalities in (22). Inserting (86) into (84) leads to

(87) ⟨ η ⟩ γ | R η ( ξ , η ) | ≤ 2 1 + γ 2 K 5,0 + 2 1 + γ 2 ∫ 0 ξ 1 T K 5,0 d t ≤ 2 2 + γ K 5,0 = K 5 .

It is easy to see that the estimate in (87) also holds for the function S _η.

For the function R _ξ(ξ, η), one utilizes (55) to obtain

(88) R ξ ( ξ , η ) = u ̃ a R η + a ( R ̃ + θ ̃ ) ( R ̃ − S ̃ ) 2 u ̃ − 1 2 f ( θ ̃ ) ( R ̃ + S ̃ + 2 θ ̃ ) + u ̃ a θ ̃ η ,

which indicates by (87) and (19) that

(89) ⟨ η ⟩ γ | R ξ ( ξ , η ) | ≤ u ̃ a ⟨ η ⟩ γ | R η | + a ⟨ η ⟩ γ + α − 2 β ⟨ η ⟩ 2 β | R ̃ + θ ̃ | ⋅ | R ̃ − S ̃ | 2 ⟨ η ⟩ α u ̃ + f ̄ 2 ⟨ η ⟩ γ − β ⋅ ⟨ η ⟩ β ( | R ̃ | + | S ̃ | + 2 | θ ̃ | ) + u ̃ a ⟨ η ⟩ γ − β ⋅ ⟨ η ⟩ β θ ̃ η ≤ K 1 a K 5 + a ( K 4 + M 1 ) ⋅ 2 K 4 2 K 0 + f ̄ 2 ( 2 K 4 + 2 M 1 ) + K 1 a M 2 = K 6 ,

which is also true for the function S _ξ(ξ, η).

Step 5. The estimates of θ _xx and θ _t. We recollect (47) and (59) to derive

(90) θ η η ( ξ , η ) = ∫ − ∞ + ∞ H ( ξ , z ) θ 0 ″ ( η − z ) d z + ∫ 0 ξ ∫ − ∞ + ∞ ∂ H ( ξ − τ , z ) ∂ z I 4 ( τ , η − z ) d z d τ ,

where

(91) I 4 = f ′ ( θ ̃ ) θ ̃ η R ̃ + S ̃ 2 + θ ̃ + f ( θ ̃ ) R ̃ η + S ̃ η 2 + θ ̃ η .

It suggests by (90) and (51) with κ = γ that

(92) ⟨ η ⟩ γ | θ η η ( ξ , η ) | ≤ 2 γ M 3,0 ∫ − ∞ + ∞ H ( ξ , z ) ( 1 + | z | γ ) d z + 2 γ f ̄ [ M 2 ( K 4 + M 1 ) + K 5 + M 2 ] ∫ 0 ξ ∫ − ∞ + ∞ ∂ H ( ξ − τ , z ) ∂ z ( 1 + | z | γ ) d z d τ ,

where the following estimate is used

(93) ⟨ η ⟩ γ | I 4 ( ξ , η ) | ≤ f ̄ [ M 2 ( K 4 + M 1 ) + K 5 + M 2 ] .

Analogous to Step 2, we apply (50) and (51) and (52) again to (92) to acquire

(94) ⟨ η ⟩ γ | θ η η ( ξ , η ) | ≤ 2 γ M 3,0 ∫ − ∞ + ∞ 1 + T γ 2 P γ H 1 ( ξ , z ) d z + 2 γ f ̄ [ M 2 ( K 4 + M 1 ) + K 5 + M 2 ] × ∫ 0 ξ ∫ − ∞ + ∞ 1 + T γ 2 P 1 + γ H 2 ( ξ − τ , z ) d z d τ ≤ 2 γ M 3,0 1 + T γ 2 P γ ⋅ 2 + 2 γ f ̄ [ M 2 ( K 4 + M 1 ) + K 5 + M 2 ] 1 + T γ 2 P 1 + γ ⋅ 4 T ≤ 2 1 + γ ( 1 + P γ ) M 3,0 + 2 2 + γ M 3,0 = 2 1 + γ ( 3 + P γ ) M 3,0 = M 3 ,

due to (19) and (22).

For the estimate of θ _t, one finds by (54) and (55), (94) and (20) that

(95) 〈 η 〉 γ | θ ξ ( ξ , η ) | ≤ 〈 η 〉 γ | θ η η ( ξ , η ) | + 〈 η 〉 γ − β | f ( θ ̃ ) | 〈 η 〉 β | R ̃ | + 〈 η 〉 β | S ̃ | 2 + 〈 η 〉 β | θ ̃ | ≤ M 3 + f ̄ ( K 4 + M 1 ) = M 4 .

Summing up (63), (65), (67), (70), (71), (80), (81), (87), (89), (94) and (95), we complete the proof of the lemma.□

Let (u, R, S, θ) be the vector function defined in Lemma 4.1. We claim that the functions (R, S, θ) are uniformly Lipschitz continuous and the function u is locally Lipschitz continuous. Indeed, it is easily seen by Lemma 4.1 that

(96) | R ( t 1 , x 1 ) − R ( t 2 , x 2 ) | + | S ( t 1 , x 1 ) − S ( t 2 , x 2 ) | + | θ ( t 1 , x 1 ) − θ ( t 2 , x 2 ) | ≤ 2 K 5 + 2 K 6 + M 2 + M 4 | t 1 − t 2 | + | x 1 − x 2 | .

For the function u, one finds by (16) that for x, y ∈ [−L, L] with a large number L ≥ 1

(97) | u 0 ( x ) − u 0 ( y ) | = ∫ y x u 0 ′ ( z ) d z ≤ K 2,0 ( 2 K 0 ) a ∫ y x 〈 z 〉 a α − β d z ≤ K 2,0 〈 L 〉 a α − β ( 2 K 0 ) a | x − y | .

Thus for any t ₁, t ₂ ∈ [0, T] and x ₁, x ₂ ∈ [−L, L], we have by (61), (63), (65) and (97)

(98) | u ( t 1 , x 1 ) − u ( t 2 , x 2 ) | ≤ | u 0 ( x ) − u 0 ( y ) | + ∫ t 2 t 1 R + S 2 + θ ( τ , x 1 ) − R + S 2 + θ ( τ , x 2 ) d τ ≤ K 2,0 ⟨ L ⟩ a α − β ( 2 K 0 ) a | x 1 − x 2 | + 2 ( K 4 + M 1 ) | t 1 − t 2 | .

We point out that, if β ≥ aα, then the function u is uniformly Lipschitz continuous.

Moreover, there has

Lemma 4.2.

The iteration mapping T defined by (62) is a contraction under the norm of L ^∞. That is, for any two vector functions ( u ̃ 1 , R ̃ 1 , S ̃ 1 , θ ̃ 1 ) ( t , x ) and ( u ̃ 2 , R ̃ 2 , S ̃ 2 , θ ̃ 2 ) ( t , x ) in Σ(T), there holds

(99) ‖ u 1 − u 2 ‖ L ∞ + ‖ R 1 − R 2 ‖ L ∞ + ‖ S 1 − S 2 ‖ L ∞ + ‖ θ 1 − θ 2 ‖ L ∞ + ‖ θ 1 x − θ 2 x ‖ L ∞ ≤ 3 4 ‖ u ̃ 1 − u ̃ 2 ‖ L ∞ + ‖ R ̃ 1 − R ̃ 2 ‖ L ∞ + ‖ S ̃ 1 − S ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 x − θ ̃ 2 x ‖ L ∞ ,

where ( u i , R i , S i , θ i ) = T ( u ̃ i , R ̃ i , S ̃ i , θ ̃ i ) for i = 1, 2.

Proof.

Since ( u ̃ i , R ̃ i , S ̃ i , θ ̃ i ) ( t , x ) ∈ Σ ( T ) , one gets by Lemma 4.1 that (u _i, R _i, S _i, θ _i) ∈ Σ(T). According to system (55), we achieve

(100) ( R 1 − R 2 ) t − u ̃ 1 a ( R 1 − R 2 ) x = F 1 + F 2 + F 3 + F 4 , ( S 1 − S 2 ) t + u ̃ 1 a ( S 1 − S 2 ) x = G 1 + G 2 + G 3 + G 4 , ( θ 1 − θ 2 ) t − ( θ 1 − θ 2 ) x x = W ,

where

(101) F 1 = a ( R ̃ 1 + θ ̃ 1 ) ( R ̃ 1 − S ̃ 1 ) 2 u ̃ 1 − a ( R ̃ 2 + θ ̃ 2 ) ( R ̃ 2 − S ̃ 2 ) 2 u ̃ 2 , G 1 = a ( S ̃ 1 + θ ̃ 1 ) ( S ̃ 1 − R ̃ 1 ) 2 u ̃ 1 − a ( S ̃ 2 + θ ̃ 2 ) ( S ̃ 2 − R ̃ 2 ) 2 u ̃ 2 , F 2 = G 2 = − 1 2 f ( θ ̃ 1 ) ( R ̃ 1 + S ̃ 1 + 2 θ ̃ 1 ) − f ( θ ̃ 2 ) ( R ̃ 2 + S ̃ 2 + 2 θ ̃ 2 ) , F 3 = − G 3 = u ̃ 1 a θ ̃ 1 x − u ̃ 2 a θ ̃ 2 x , F 4 = u ̃ 1 a − u ̃ 2 a R 2 x , G 4 = u ̃ 2 a − u ̃ 1 a S 2 x , W = f ( θ ̃ 1 ) R ̃ 1 + S ̃ 1 2 + θ ̃ 1 − f ( θ ̃ 2 ) R ̃ 2 + S ̃ 2 2 + θ ̃ 2 .

One performs direct calculations by (54) to conclude that

(102) | F 1 | , | G 1 | ≤ a K 3 K 0 ‖ R ̃ 1 − R ̃ 2 ‖ L ∞ + ‖ S ̃ 1 − S ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ + a K 3 K 4 K 0 2 ‖ u ̃ 1 − u ̃ 2 ‖ L ∞ , | F 2 | , | G 2 | ≤ f ̄ K 3 ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ + f ̄ ‖ R ̃ 1 − R ̃ 2 ‖ L ∞ + ‖ S ̃ 1 − S ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ , | F 3 | , | G 3 | ≤ K 1 a ‖ θ ̃ 1 x − θ ̃ 2 x ‖ L ∞ + a M 2 K 1 a − 1 ‖ u ̃ 1 − u ̃ 2 ‖ L ∞ , | F 4 | , | G 4 | ≤ a K 1 a − 1 K 5 ‖ u ̃ 1 − u ̃ 2 ‖ L ∞ , | W | ≤ f ̄ K 3 ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ + f ̄ ‖ R ̃ 1 − R ̃ 2 ‖ L ∞ + ‖ S ̃ 1 − S ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ .

Putting (102) into (100) and noting the initial data (R ₁ − R ₂)(0, x) = (S ₁ − S ₂)(0, x) = (θ ₁ − θ ₂)(0, x) = 0, we have

(103) ‖ R 1 − R 2 ‖ L ∞ , ‖ S 1 − S 2 ‖ L ∞ ≤ K ̂ T ‖ R ̃ 1 − R ̃ 2 ‖ L ∞ + ‖ S ̃ 1 − S ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ + ‖ u ̃ 1 − u ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 x − θ ̃ 2 x ‖ L ∞ ,

where

(104) K ̂ = a K 3 K 0 + a K 3 K 4 K 0 2 + 2 f ̄ ( K 3 + 1 ) + K 1 a + a M 2 K 1 a − 1 + a K 1 a − 1 K 5 ,

and

(105) ‖ θ 1 − θ 2 ‖ L ∞ ≤ ∫ 0 T ∫ − ∞ + ∞ H ( t − τ , z ) | W | d τ d z ≤ 2 f ̄ ( K 3 + 1 ) T ‖ R ̃ 1 − R ̃ 2 ‖ L ∞ + ‖ S ̃ 1 − S ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ .

Furthermore, it follows by (47) and (100) that

(106) ‖ θ 1 x − θ 2 x ‖ L ∞ ≤ ∫ 0 T ∫ − ∞ + ∞ ∂ H ( t − τ , z ) ∂ z ⋅ | W | d τ d z ≤ 4 f ̄ ( K 3 + 1 ) T ‖ R ̃ 1 − R ̃ 2 ‖ L ∞ + ‖ S ̃ 1 − S ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ .

The estimate |u ₁ − u ₂| can be obtained by (61), (103) and (105)

(107) ‖ u 1 − u 2 ‖ L ∞ ≤ T ‖ R 1 − R 2 ‖ L ∞ + ‖ S 1 − S 2 ‖ L ∞ + ‖ θ 1 − θ 2 ‖ L ∞ ≤ 3 K ̂ T 2 ‖ R ̃ 1 − R ̃ 2 ‖ L ∞ + ‖ S ̃ 1 − S ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 − θ ̃ 2 ‖ L ∞ + ‖ u ̃ 1 − u ̃ 2 ‖ L ∞ + ‖ θ ̃ 1 x − θ ̃ 2 x ‖ L ∞ .

We add (103) and (105)–(107) and use the facts K ̂ T ≤ 1 12 , f ̄ ( K 3 + 1 ) T ≤ 1 16 to obtain (99). The proof of the lemma is ended.□

4.3 The existence and uniqueness of solutions

In this subsection, we complete the proof of Theorem 1. There has

Lemma 4.3.

Suppose that the conditions in Theorem 1 hold. Then there exists a unique vector function (u, R, S, θ) ∈ Σ(T) and x _±(t) = x _±(t; ξ, η) for any point ( ξ , η ) ∈ [ 0 , T ] × R such that

(108) R ( ξ , η ) = R 0 ( x − ( 0 ) ) + ∫ 0 ξ a ( R + θ ) ( R − S ) 2 u − 1 2 f ( θ ) ( R + S + 2 θ ) + u a θ x ( t , x − ( t ) ) d t , S ( ξ , η ) = S 0 ( x + ( 0 ) ) + ∫ 0 ξ a ( S + θ ) ( S − R ) 2 u − 1 2 f ( θ ) ( R + S + 2 θ ) − u a θ x ( t , x + ( t ) ) d t ,

and

(109) θ ( ξ , η ) = ∫ − ∞ + ∞ H ( ξ , z ) θ 0 ( η − z ) d z + ∫ 0 ξ ∫ − ∞ + ∞ H ( ξ − τ , z ) f ( θ ) R + S 2 + θ ( τ , η − z ) d z d τ , θ η ( ξ , η ) = ∫ − ∞ + ∞ H ( ξ , z ) θ 0 ′ ( η − z ) d z + ∫ 0 ξ ∫ − ∞ + ∞ ∂ H ( ξ − τ , z ) ∂ z f ( θ ) R + S 2 + θ ( τ , η − z ) d z d τ , u ( ξ , η ) = u 0 ( η ) + ∫ 0 ξ R + S 2 + θ ( τ , η ) d τ ,

and

(110) x ± ( t ; ξ , η ) = η ± ∫ ξ t u a ( τ , x ± ( τ ; ξ , η ) ) d τ .

Proof.

The proof of the lemma is based on Lemmas 4.1, 4.2 and the Arzela–Ascoli Theorem. Denote (u ⁽⁰⁾, R ⁽⁰⁾, S ⁽⁰⁾, θ ⁽⁰⁾) = (u ₀, R ₀, S ₀, θ ₀) ∈ Σ(T). Thanks to the mapping T defined by (62), we can construct an iteration sequence for n ≥ 0

(111) ( u ( n + 1 ) , R ( n + 1 ) , S ( n + 1 ) , θ ( n + 1 ) ) = T ( u ( n ) , R ( n ) , S ( n ) , θ ( n ) ) .

On account of Lemmas 4.1 and 4.2, it is known that (u ⁽ⁿ⁾, R ⁽ⁿ⁾, S ⁽ⁿ⁾, θ ⁽ⁿ⁾) ∈ Σ(T) for each n ≥ 1 and the iteration sequence { ( u ( n ) , R ( n ) , S ( n ) , θ ( n ) ) } n = 0 ∞ converges a fixed point (u, R, S, θ) in the norm of L ^∞. Moreover, making use of the sequence { u ( n ) } n = 0 ∞ , one can uniquely define a sequence of characteristic curves x ± ( n ) ( t ; ξ , η ) n = 0 ∞ for any point (ξ, η) ∈ [0, T], because u ⁽ⁿ⁾(n ≥ 0) are bounded and local Lipschitz continuous by (98). Furthermore, for arbitrarily fixed number L ≥ 1, we find that the curve sequence x ± ( n ) ( t ; ξ , η ) n = 0 ∞ is uniform bounded and uniform equicontinuous. Therefore, due to the Arzela–Ascoli Theorem, there exists a subsequence of x ± ( n ) ( t ; ξ , η ) n = 0 ∞ , still denoted by the same notation for convenience, such that x ± ( n ) ( t ; ξ , η ) converges uniformly to a smooth curve x _±(t; ξ, η) on [0, T] × [0, T] × [−L, L]. It is noted that this choice of the subsequence is depending on the number L. However, a subsequence which is independent of L can be reselected by the Cantor’s diagonal argument such that the convergence of the subsequence is valid on [0, T] × [0, T] × [−L′, L′] for any L′ ≥ 1. Utilizing (96) and (98) again yields

(112) ( u ( n ) , R ( n ) , S ( n ) , θ ( n ) ) ( t , x ± ( n ) ( t ) ) → ( u , R , S , θ ) ( t , x ± ( t ) ) ,

and θ x ( n ) ( t , x ± ( n ) ( t ) ) → θ x ( t , x ± ( t ) ) as n → ∞, which indicate that (108) and (110) hold. It is directly seen that the relations in (109) are satisfied.

Next we check that the limit vector function (u, R, S, θ) is in Σ(T). According to the expressions in (108) and (109), we easily obtain the estimates for R, S, θ, θ _x, u and u _t

(113) | R | , | S | ≤ K 4 ⟨ x ⟩ − β , | θ | ≤ M 1 ⟨ x ⟩ − β , | θ x | ≤ M 2 ⟨ x ⟩ − β , K 0 ⟨ x ⟩ − α ≤ u ( t , x ) ≤ K 1 , | u t | ≤ K 3 ⟨ x ⟩ − β .

Here u _t is achieved by differentiating the equation for u in (109) with respect to ξ

(114) u ξ ( ξ , η ) = R + S 2 + θ .

Since the determination of the constant K ₂ is independent of the functions (R _x, S _x, R _t, S _t), one can perform the same procedure as in Step 3 in Subsection 4.2 to derive the estimate of the function u _x

(115) u a | u x | ≤ K 2 〈 x 〉 − β , o r | u x | ≤ K 2 K 0 a 〈 x 〉 a α − β .

In order to estimate the functions R _x, S _x and θ _x, we differentiate (108) and (109) with respect to η to deduce

(116) R η ( ξ , η ) = R 0 ′ ( x − ( 0 ) ) ∂ x − ( 0 ) ∂ η + ∫ 0 ξ I 5 ( t , x − ( t ) ) ∂ x − ( t ) ∂ η d t , S η ( ξ , η ) = S 0 ′ ( x + ( 0 ) ) ∂ x + ( 0 ) ∂ η + ∫ 0 ξ I 6 ( t , x + ( t ) ) ∂ x + ( t ) ∂ η d t , θ η η ( ξ , η ) = ∫ − ∞ + ∞ H ( ξ , z ) θ 0 ″ ( η − z ) d z + ∫ 0 ξ ∫ − ∞ + ∞ ∂ H ( ξ − τ , z ) ∂ z I 7 ( τ , η − z ) d z d τ ,

where I ₅ = I _5,1 R _x + I _5,2 S _x + u ^a θ _xx + I _5,3, I ₆ = I _6,1 S _x + I _6,2 R _x − u ^a θ _xx + I _6,3 and I ₇ = I _7,1 R _η + I _7,2 S _η + I _7,3. The coefficients in I _5,6,7 are

I 5,1 = a ( 2 R − S + θ ) 2 u − f ( θ ) 2 , I 5,2 = − a ( R + θ ) 2 u − f ( θ ) 2 , I 5,3 = a ( R − S ) 2 u − 1 2 f ′ ( θ ) ( R + S + 2 θ ) − f ( θ ) + a u a − 1 u x θ x − a ( R + θ ) ( R − S ) 2 u 2 u x , I 6,1 = a ( 2 S − R + θ ) 2 u − f ( θ ) 2 , I 6,2 = − a ( S + θ ) 2 u − f ( θ ) 2 , I 6,3 = a ( S − R ) 2 u − 1 2 f ′ ( θ ) ( R + S + 2 θ ) − f ( θ ) − a u a − 1 u x θ x − a ( S + θ ) ( S − R ) 2 u 2 u x , I 7,1 = I 7,2 = f ( θ ) 2 , I 7,3 = f ′ ( θ ) R + S 2 + f ′ ( θ ) θ + f ( θ ) θ η .

We use the same process as in the proof of (87) and (94) to establish the boundedness of ⟨η⟩^γ R _η, ⟨η⟩^γ S _η and ⟨η⟩^γ θ _ηη. To estimate the functions R _t, S _t and θ _t, one differentiates (108) and (109) with respect to ξ to get the equations of (R _ξ, S _ξ, θ _ξ)(ξ, η). One can adopt the similar treatments as in (89) and (95) to acquire the estimates of the functions R _t, S _t and θ _t. Hence we have verified (u, R, S, θ) ∈ Σ(T). In addition, the system (114) also imply that the vector function (u, R, S, θ) satisfy the differential system (27). Thus (u, R, S, θ) is a classical solution to the Cauchy problem (27), (28).

Finally, we show that the classical solution of the Cauchy problem (27), (28) is unique. Let (u ₁, R ₁, S ₁, θ ₁) and (u ₂, R ₂, S ₂, θ ₂) ∈ Σ(T) be two solution of the Cauchy problem (27), (28). Denote ( u ̂ , R ̂ , S ̂ , θ ̂ ) = ( u 1 − u 2 , R 1 − R 2 , S 1 − S 2 , θ 1 − θ 2 ) . Analogous to that of (100), the functions ( u ̂ , R ̂ , S ̂ , θ ̂ ) satisfy the following system

(117) R ̂ t − u 1 a R ̂ x = F ̂ 1 + F ̂ 2 + F ̂ 3 + F ̂ 4 , S ̂ t + u 1 a S ̂ x = G ̂ 1 + G ̂ 2 + G ̂ 3 + G ̂ 4 , θ ̂ t − θ ̂ x x = W ̂ , u ̂ t = R ̂ + S ̂ 2 + θ ̂ ,

where F ̂ i , G ̂ i and W ̂ are given in (101) but with (u _i, R _i, S _i, θ _i) replacing ( u ̃ i , R ̃ i , S ̃ i , θ ̃ i ) for i = 1, 2. We perform similar arguments as in Lemma 4.2 to conclude ‖ u ̂ ‖ L ∞ = ‖ R ̂ ‖ L ∞ = ‖ S ̂ ‖ L ∞ = ‖ θ ̂ ‖ L ∞ = 0 . The proof of the lemma is completed.□

It is evident to see that the existence and uniqueness of classical solutions to the Cauchy problem (27), (28) is established by Lemma 4.3. We claim that the two Cauchy problems (27), (28) and (1), (2) are equivalent, which completes the proof of Theorem 1. It suffices to show that there holds

(118) u x = R − S 2 u a ,

for any ( t , x ) ∈ [ 0 , T ] × R . One first recalls (29) to find

(119) u x | t = 0 = u 0 ′ ( x ) = R 0 − S 0 2 u 0 a = R − S 2 u a t = 0 ,

which indicates that (118) is true for t = 0. For t > 0, we differentiate the equation for u in (109) with respect to the spatial variable and utilize (27) to gain

(120) u x = u 0 ′ + ∫ 0 t R x + S x 2 + θ x ( τ , x ) d τ = u 0 ′ + ∫ 0 t 1 2 u a R τ − S τ − a ( R + S + 2 θ ) ( R − S ) 2 u − 2 u a θ x + θ x ( τ , x ) d τ = u 0 ′ + ∫ 0 t R τ − S τ 2 u a − a u τ ( R − S ) 2 u a + 1 ( τ , x ) d τ ,

from which and the integration by parts, one has by (119)

(121) u x = u 0 ′ + R − S 2 u a τ = 0 τ = t − ∫ 0 t ( R − S ) 1 2 u a τ d τ − ∫ 0 t a u τ ( R − S ) 2 u a + 1 ( τ , x ) d τ = u 0 ′ + R − S 2 u a − R 0 − S 0 2 u 0 a = R − S 2 u a ,

which is the desired equation (118). Therefore we complete the proof of Theorem 1.

Corresponding author: Yuusuke Sugiyama, School of Engineering, University of Shiga Prefecture, 5228533, Hikone-City, Shiga, Japan, E-mail: sugiyama.y@e.usp.ac.jp

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 12171130

Award Identifier / Grant number: 12071106

Funding source: Grants-in-Aid for Scientific Research

Award Identifier / Grant number: 23K03169

Acknowledgements

The authors wish to thank the anonymous referees for their careful reading of the paper and useful comments.

Research ethics: Not applicable.
Author contributions: The authors has accepted responsibility for the entire content of this manuscript and approved its submission.
Competing interests: The authors states no conflict of interest.
Research funding: Y. Hu was partially supported by the National Natural Science Foundation of China (Nos. 12171130 and 12071106). Y. Sugiyama was partially supported by Grants-in-Aid for Scientific Research (C) (No. 23K03169).
Data availability: Not applicable.

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Received: 2023-09-12

Accepted: 2024-05-20

Published Online: 2024-06-17

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https://doi.org/10.1515/ans-2023-0142

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thermoelasticity system; well-posedness; classical solvability; degeneracy

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