Existence of global solutions to a semilinear thermoelastic system in three dimensions

Yaqing Sun; Daoyin He; Kangqun Zhang

doi:10.1515/math-2025-0205

Article Open Access

Existence of global solutions to a semilinear thermoelastic system in three dimensions

Yaqing Sun , Daoyin He and Kangqun Zhang

Published/Copyright: November 24, 2025

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$Open Mathematics$

From the journal Open Mathematics Volume 23 Issue 1

Abstract

In this paper, we establish the global existence of weak solutions to a semilinear thermoelastic system on a bounded domain in R 3 . Specifically, we assume that the nonlinear term in the momentum equation is defocusing energy critical. The proofs are based on the existence of the strong solutions of linear thermoelastic system and the method of energy estimates from wave equation.

Keywords: 3-D thermoelastic system; defocusing; energy critical; approximation; global weak solutions

MSC 2020: 35M30; 35M33

1 Introduction

We are concerned with the global existence of weak solutions to a 3-D semilinear thermoelastic system. More specifically, we consider the following initial-boundary value problem:

(1) ∂ t 2 u − Δ u + b ∇ θ = g ( u ) in Ω × ( 0 , + ∞ ) ,

(2) ∂ t θ − a Δ θ + b ∂ t div u = h ( θ ) in Ω × ( 0 , + ∞ ) ,

(3) u ( x , t ) = 0 , θ ( x , t ) = 0 on ∂ Ω × ( 0 , + ∞ ) ,

(4) u ( x , 0 ) = u ( 0 ) , ∂ t u ( x , 0 ) = u ( 1 ) , θ ( x , 0 ) = θ ( 0 ) in Ω ,

where Ω is a bounded connected domain in R 3 with boundary ∂Ω ∈ C ². Here a > 0 and b ≠ 0 are constants, u = (u ₁, u ₂, u ₃) and θ denote the displacement and the temperature, respectively.

For the momentum equation (1), let g ( u ) = − u 1 5 , − u 2 5 , − u 3 5 , then for j = 1, 2, 3, u _j satisfies

(5) ∂ t 2 u j − Δ u j + b ∂ x j θ + u j 5 = 0 ,

which is a defocusing energy critical wave equation. Indeed, for each fixed j, ∫ 0 u j s 5 d s ≥ 0 , thus the energy is defocusing; the index 5 is the critical index of 3-D semilinear wave equation.

For the nonlinear term in the energy equation (2), due to technical reason in energy estimate, we assume that h ∈ C ¹ and h(0) = 0, furthermore, we introduce the following condition

(6) h ( θ ) θ ≤ C θ 2 ,

where C > 0 is a universal constant. Without loss of generality, we might as well set the following constraint

(7) Ω ⊆ B ( 0,1 ) = { x ∈ R 3 : | x | ≤ 1 } .

Now we can state the main result of this paper:

Theorem 1.1.

Assume that the initial data u ( 0 ) ∈ H 0 1 ( Ω ) , u ₍₁₎ ∈ L ²(Ω), θ ( 0 ) ∈ H 0 1 ( Ω ) , and (6) holds. Then the system (1)– (4) admits a global weak solution (u, θ) such that

(8) u ∈ C [ 0 , ∞ ) , H 0 1 ( Ω ) ∩ C 1 [ 0 , ∞ ) , L 2 ( Ω ) ,

(9) θ ∈ L 2 [ 0 , ∞ ) , H 0 1 ( Ω ) ∩ C [ 0 , ∞ ) , L 2 ( Ω ) .

To place our result in context, we review a few highlights from thermoelastic systems. As we all know, the thermoelastic system is a hyperbolic-parabolic coupled system which describes the elastic and the thermal behavior. The pioneering work of the thermoelastic system could date back to C. Dafermos [1] on linear thermoelasticity. For the linear case, the rich amount of literature produced since then is concerned with global existence, uniqueness and stability of linear problem. J. U. Kim [2] showed that the energy of a thermoelastic bar and plate decays exponentially fast. A. E. Green and P. M. Naghdi [3] proposed three models, based on the different material responses, labeled as types I, II and III. The linearized version of the first model corresponds to the Fourier law, the linearized version of both types II and III models whose constitutive assumptions on the heat flux vector are different from the Fourier’s law allows heat transmission at a finite speed. For the thermoelasticity of type I, M. Slemrod [4] proved the global existence, uniqueness and asymptotic stability of classical smooth solutions of the linear system. For the thermoelasticity of type II, or no energy dissipation exists, Y. Qin, S. Deng and L. Huang et al. [5] obtained the global existence for the three-dimensional thermoelastic equations of type II. B. Lazzari and R. Nibbi [6] obtained the exponential decay of total energy to thermoelastic linear inhomogeneous system of type II. For the thermoelasticity of type III, which presents thermal dissipation, there are some interesting results, for example, for the Cauchy problem of the linear thermoelastic system of type III, X. Zhang and E. Zuazua [7], and R. Quintanilla and R. Racke [8] independently studied the decay of energy by using the classical energy method and the spectral method, and obtained the exponential stability in one space dimension, and in two or three space dimensions for radially symmetric situations, while the energy decays polynomially for most domains in two space dimensions. M. Reissig and Y. Wang [9] studied the L ^p − L ^q decays estimates and propagation of singularities of solutions in one space dimension, and later on L. Yang and Y. Wang [10] studied well-posedness and decay estimates in three space dimension.

For the nonlinear case, there are few results and most of which require the global Lipschitz condition for the nonlinear term. H. Gao and J. E. Muñoz Rivera [11] established global existence and decay for the semilinear thermoelastic contact problem in the one dimensional case. For 3D semilinear thermoelastical system with general nonlinear term, Y. Wang [12] introduced a linear transformation to decouple the hyperbolic operator and parabolic operator, by using the decoupled system and techniques from microlocal analysis, he managed to establish local existence and uniqueness for the strong solution. However, due to the lack of structural condition, global existence was unknown in [12]. For the nonlinear thermoelastic system with random structure, H. Gao in [13] got existence of global attractor for semilinear thermoelastic problem. T. Caraballo, I. Chueshov and J. A. Langa [14] obtained existence of invariant manifolds for certain stochastic thermoelastic system. Recently, F. D. M. Bezerra and V. L. Nascimento [15] study a class of semilinear thermoelastic systems with variable thermal coefficient, they establish the existence, regularity and upper semicontinuity of the pullback attractors with respect to the coefficients of thermal expansion of the material.

In this paper, the nonlinear terms on the right hand side of the momentum equation (1) correspond to the defocusing energy critical case. The defocusing energy critical problem is a very important topic in wave equation, the 3-D case was first solved by Grillakis [16], one can also consult [17]. Moreover, note that in most of the former works of thermoelasticity in nonlinear case, the nonlinear terms were restricted to satisfy global Lipschitz condition, see for instance [11], [13] and [14]. However, in our paper, both the defocusing energy critical nonlinear term g(u) in (1) and the nonlinear term h(θ) in (2) do not satisfy global Lipschitz condition, thus it is difficult to apply the contraction mapping theory in the proof of global existence. In order to overcome this difficulty, motivated by Struwe [17], we approximate the nonlinear terms with global Lipschitz functions and utilize the structure of assumption (6), so that the contraction mapping theory is applicable. Finally, we get the global existence of weak solutions. Also, we mention that, due to the generality of nonlinear terms, Y. Wang only got the local existence result in [12]. If we restrict us to the local well-posedness of (1)– (4), then we are able to deal with nonlinear terms which only satisfying local Lipschitz condition, and the local existence and uniqueness of strong solution with more regular initial data can be established via an alternative approach, see Section 3.2 for details.

Now we comment on the proof of Theorem 1.1. Motivated by [17], we use the method of energy estimate. Firstly, for the 3-D linear thermoelastic problem, we get global existence of strong solutions by semigroup method and establish the necessary energy estimates. Next, for system (1)– (4), we approximate the nonlinear terms (g, h) with a series of global Lipschitz functions { ( g ( k ) , h ( k ) ) } k ∈ N , then for each k, based on the result of linear case, we get a unique global weak solution (u ^(k), θ ^(k)) for the corresponding approximating initial-boundary value problem. To proceed further, we need to improve the regularity of (u ^(k), θ ^(k)), such that u , θ ∈ C ( [ 0 , T ] ; H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) . For this aim, we emphasize that taking difference quotients in x variable wouldn’t help us during the proof since we are dealing with an initial boundary problem. To overcome this difficulty, we take difference quotients in t variable instead, then consider the system of (∂ _t u ^(k), ∂ _t θ ^(k)) and establish the L ² space estimates of ∂ t 2 u ( k ) and ∂ _t θ ^(k). Hence the H ² space estimates for (u ^(k), θ ^(k)) follow by solving the elliptic equations, and (u ^(k), θ ^(k)) turns out to be the unique strong solution for the corresponding approximating initial-boundary value problem. Finally, we let k → ∞ to get the global weak solution for the original system (1)– (4). Since g admits an energy critical index, the uniqueness is unknown. The uniqueness and higher regularity of the global solution will be given in our next paper, we will also consider (1)– (4) for unbounded domain Ω in our future work.

The remaining part of this paper is organized as follows: In Section 2, we give the existence of linear problem and establish the necessary energy estimate. In Section 3, we prove Theorem 1.1, finally, we obtain the existence and uniqueness of the local strong solution.

2 Global existence for linear system

Before we solve the nonlinear system (1)– (4), we firstly consider the linear case:

(10) ∂ t 2 u − Δ u + b ∇ θ = g ( t , x ) in Ω × ( 0 , + ∞ ) ,

(11) ∂ t θ − a Δ θ + b ∂ t div u = h ( t , x ) in Ω × ( 0 , + ∞ ) ,

(12) u ( x , t ) = 0 , θ ( x , t ) = 0 on ∂ Ω × ( 0 , + ∞ ) ,

(13) u ( x , 0 ) = u ( 0 ) , ∂ t u ( x , 0 ) = u ( 1 ) , θ ( x , 0 ) = θ ( 0 ) , in Ω .

Here, we assume that g , h ∈ C 1 ( [ 0 , ∞ ) , L 2 ( Ω ) and u ( 0 ) ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) , u ( 1 ) ∈ H 0 1 ( Ω ) , θ ( 0 ) ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) .

Next we will apply the semigroup method. For this aim, we introduce a new variable v = ∂ _t u and let

U = u v θ .

Then the system (10)– (13) is converted into the following first-order system

(14) ∂ t U + A U = F ,

where

A U = 0 − I 3 0 − I 3 Δ 0 b ∇ x 0 b ∇ x T − a Δ u v θ = − v − Δ u + b ∇ x θ b div v − a Δ θ ,

and

F = 0 g h .

Definition 2.1.

We say (u, θ) is a strong solution of (10)– (13) if (u, θ) satisfy (10)– (13) in the sense of distribution and satisfies the following regularity:

(15) u ∈ C [ 0 , ∞ ) , H 2 ( Ω ) ∩ H 0 1 ( Ω ) ⋂ C 1 [ 0 , ∞ ) , H 0 1 ( Ω ) ,

(16) θ ∈ C [ 0 , ∞ ) , H 2 ( Ω ) ∩ H 0 1 ( Ω ) ⋂ C 1 [ 0 , ∞ ) , L 2 ( Ω ) .

With Definition 2.1, we will concern about system (14) in the space

(17) D ( A ) = H 2 ( Ω ) ∩ H 0 1 ( Ω ) × H 0 1 ( Ω ) × H 2 ( Ω ) ∩ H 0 1 ( Ω ) ,

then one can easily check that the operator A is a maximal accretive operator, thus by Theorem 2.4.1 of [18], we have the following lemma:

Lemma 2.2.

If the initial data (u ₍₀₎, u ₍₁₎, θ ₍₀₎) ∈ D(A), then the linear system (10)– (13) admits a unique strong solution (u, θ).

For system (10)– (13), a direct computation yields the following energy equality:

(18) d d t ∫ Ω | ∂ t u | 2 + | ∇ u | 2 + | θ | 2 d x + 2 a ∫ Ω | ∇ θ | 2 d x = 2 ∫ Ω g ∂ t u d x + 2 ∫ Ω h θ d x ,

then we can define the energy function

(19) E 0 u ( t ) , θ ( t ) = ∫ Ω | ∂ t u | 2 + | ∇ u | 2 + | θ | 2 ( ⋅ , t ) d x .

Combined with Hölder’s inequality, we derive

d d t E 0 u ( t ) , θ ( t ) ≤ 2 E 0 u ( t ) , θ ( t ) 1 2 ( ‖ g ( ⋅ , t ) ‖ L 2 ( Ω ) + ‖ h ( ⋅ , t ) ‖ L 2 ( Ω ) ) .

Thus by denoting

(20) ‖ ( u , θ ) ( t ) ‖ 0 : = E 0 u ( t ) , θ ( t ) 1 2

we have

Corollary 2.3.

The global solutions of system (10)– (13) satisfy

(21) d d t ‖ ( u , θ ) ( t ) ‖ 0 ≤ ‖ g ( ⋅ , t ) ‖ L 2 ( Ω ) + ‖ h ( ⋅ , t ) ‖ L 2 ( Ω ) .

3 Existence of nonlinear system

In this section, we consider the original system (1)– (4) with condition (6). Recall that in [11], [13] and [14], the nonlinear terms were required to satisfy the global Lipschitz condition, then the contraction map principle is applicable. However, for our case, the nonlinear terms g(u) and h(θ) do not admit the global Lipschitz condition. To overcome this difficulty, motivated by the idea in [17], we approximate the nonlinear terms by global Lipschitz functions, then use the solvability of the approximate systems together with the energy estimates to construct a global weak solution for system (1)– (4). However, due to the energy critical index of g(u), we cannot get the uniqueness. On the other hand, if we assume higher regularity of the initial data and apply semigroup method, we can obtain an unique local strong solution. This result will be presented after the proof of Theorem 1.1.

3.1 Global existence of weak solutions

We will prove Theorem 1.1 in the following two steps:

Step (i) Global strong solutions for global Lipschitz nonlinear terms

As we stated above, firstly, we assume that g(u) and h(θ) are global Lipschitz, then establish the global existence result and necessary energy estimates for the corresponding thermoelastic system. More precisely, we assume that there exists a constant L > 0, such that

(22) | g ( u ) − g ( u ̃ ) | ≤ L | u − u ̃ | , | h ( θ ) − h ( θ ̃ ) | ≤ L | θ − θ ̃ | .

We also assume that

(23) g ( 0 ) = 0 , h ( 0 ) = 0 .

We mention that one cannot get enough regularity of the solutions if one applies directly the theory of semigroup as in [18]. To handle this, we use the energy method. For D(A) defined in (17), we fix initial data (u ₍₀₎, u ₍₁₎, θ ₍₀₎) ∈ D(A). Then for any functions (w, ϕ) such that w ∈ C [ 0 , ∞ ) , H 0 1 ( Ω ) ∩ C 1 [ 0 , ∞ ) , L 2 ( Ω ) , ϕ ∈ C [ 0 , ∞ ) , H 0 1 ( Ω ) ∩ C 1 [ 0 , ∞ ) , L 2 ( Ω ) , we consider the following initial-boundary value problem

(24) ∂ t 2 u − Δ u + b ∇ x θ = g ( w ) in Ω × ( 0 , + ∞ ) , ∂ t θ − a Δ θ + b ∂ t div u = h ( ϕ ) in Ω × ( 0 , + ∞ ) , u ( x , t ) = 0 , θ ( x , t ) = 0 on ∂ Ω × ( 0 , + ∞ ) , u ( x , 0 ) = u ( 0 ) , ∂ t u ( x , 0 ) = u ( 1 ) , θ ( x , 0 ) = θ ( 0 ) , on Ω .

Combined with (22)– (23), g ( w ) , h ( ϕ ) ∈ C 1 [ 0 , ∞ ) , L 2 ( Ω ) , thus by Lemma 2.2, we obtain a solution (u, θ). Since the initial data (u ₍₀₎, u ₍₁₎, θ ₍₀₎) are fixed, we may denote (u, θ) = K(w, ϕ). Further more, by the energy estimate (21), we have for w , ϕ , w ̃ , ϕ ̃ ∈ C [ 0 , ∞ ) , H 0 1 ( Ω ) ∩ C 1 [ 0 , ∞ ) , L 2 ( Ω ) and 0 < T ≤ 1,

sup 0 ≤ t ≤ T K ( w , ϕ ) − K ( w ̃ , ϕ ̃ ) ( t ) 0 ≤ ∫ 0 T g ( w ) − g ( w ̃ ) ( t ) L 2 ( Ω ) + h ( ϕ ) − h ( ϕ ̃ ) ( t ) L 2 ( Ω ) d t ≤ L ∫ 0 T w − w ̃ ( t ) L 2 ( Ω ) + ϕ − ϕ ̃ ( t ) L 2 ( Ω ) d t ,

by Poincáre’s inequality,

(25) sup 0 ≤ t ≤ T K ( w , ϕ ) − K ( w ̃ , ϕ ̃ ) ( t ) 0 ≤ L T sup 0 ≤ t ≤ T 1 + t 2 ∇ w − w ̃ ( t ) L 2 ( Ω ) + ϕ − ϕ ̃ ( t ) L 2 ( Ω ) ≤ 2 L T sup 0 ≤ t ≤ T ‖ ( w − w ̃ , ϕ − ϕ ̃ ) ‖ 0 .

Then, for T ≤ min { 1 , 1 4 L } , K can be extended to a contraction map on the space:

V = u ∈ C ( [ 0 , T ] ; H 0 1 ( Ω ) ) , θ ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) ∣ ∂ t u ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) , sup 0 ≤ t ≤ T ‖ ( u , θ ) ( t ) ‖ 0 < ∞ .

Thus, we get a unique local weak solution (u, θ). Furthermore, one also obtain from (18) that for 0 ≤ t ≤ T

(26) a ∫ 0 t ∫ Ω | ∇ θ | 2 d x d τ ≤ ‖ ( u , θ ) ( 0 ) ‖ 0 2 + ∫ 0 t ∫ Ω g ( u ) ∂ t u + h ( θ ) θ d x d τ ≤ ‖ ( u , θ ) ( 0 ) ‖ 0 2 + ∫ 0 t ∫ Ω L | u ‖ ∂ t u | + | θ | 2 d x d τ ≤ ‖ ( u , θ ) ( 0 ) ‖ 0 2 + L ∫ 0 t ∫ Ω | ∂ t u | 2 + | u | 2 + | θ | 2 d x d τ ≤ ‖ ( u , θ ) ( 0 ) ‖ 0 2 + L ( T + 1 ) ∫ 0 t ‖ ( u , θ ) ( τ ) ‖ 0 2 d τ < ∞ .

Thus ( u , ∂ t u , θ ) ∈ C ( [ 0 , T ] ; H 0 1 ( Ω ) ) × C ( [ 0 , T ] ; L 2 ( Ω ) ) × L 2 ( [ 0 , T ] ; H 0 1 ( Ω ) ) . For any T ₀ > T, repeating the procedure above for finite times, we can get a unique weak solution for 0 ≤ t ≤ T ₀, since T ₀ could be arbitrarily large, global existence and uniqueness for the global Lipschitz case is established.

In the next step, we will improve the space regularity of the global weak solution (u, θ). We shall first improve the regularity in time variable, then the regularity in space variables can be improved by solving the elliptic equations of (u, θ). To begin with, taking difference quotients

u t ( s ) = u ( ⋅ + s , x ) − u ( ⋅ , x ) s , θ t ( s ) = θ ( ⋅ + s , x ) − θ ( ⋅ , x ) s

in time variable and letting s → 0, we get that (∂ _t u, ∂ _t θ) is a weak solution of the following system

(27) ∂ t 2 ( ∂ t u ) − Δ ( ∂ t u ) + b ∇ x ( ∂ t θ ) = ∇ u g ( u ) ∂ t u in Ω × ( 0 , + ∞ ) , ∂ t ( ∂ t θ ) − a Δ ( ∂ t θ ) + b ∂ t div ( ∂ t u ) = h ′ ( θ ) ∂ t θ in Ω × ( 0 , + ∞ ) , ∂ t u ( x , t ) = 0 , ∂ t θ ( x , t ) = 0 on ∂ Ω × ( 0 , + ∞ ) , ∂ t u ( x , 0 ) = u ( 1 ) , ∂ t 2 u ( x , 0 ) = Δ u ( 0 ) − b ∇ θ 0 + g ( u ( 0 ) ) , ∂ t θ ( x , 0 ) = a Δ θ ( 0 ) − b div u ( 1 ) + h ( θ ( 0 ) ) .

Multiply the first equation in (27) by ∂ t 2 u and the second one by ∂ _t θ, then sum them up to get

1 2 d d t | ∂ t 2 u | 2 + | ∇ ∂ t u | 2 + | ∂ t θ | 2 − div ∂ t 2 u ∇ ∂ t u + a ∂ t θ ∇ ∂ t θ + a | ∇ ∂ t θ | 2 + b ∂ t 2 u ⋅ ∇ ∂ t θ + ∂ t θ div ∂ t 2 u = ∇ u g ( u ) ∂ t u ⋅ ∂ t 2 u + h ′ ( θ ) ∂ t θ ∂ t θ .

Then we get

‖ ∂ t u , ∂ t θ ( t ) ‖ 0 ≤ ‖ ∂ t u , ∂ t θ ( 0 ) ‖ 0 + ∫ 0 t ∇ u g ( u ) ∂ t u ( τ ) L 2 ( Ω ) + h ′ ( θ ) ∂ t θ ( τ ) L 2 ( Ω ) d τ

By our assumption (22), we have that

| ∇ u g ( u ) | ≤ L , | h ′ ( θ ) | ≤ L .

This observation together with Poincáre’s inequality yields for 0 < t ≤ T

(28) ‖ ∂ t u , ∂ t θ ( t ) ‖ 0 ≤ ‖ ∂ t u , ∂ t θ ( 0 ) ‖ 0 + 2 L ( T + 1 ) ∫ 0 t ‖ ∂ t u , ∂ t θ ( τ ) ‖ 0 d τ .

By Gronwall’s inequality, for any T > 0

(29) sup 0 ≤ t ≤ T ‖ ∂ t u , ∂ t θ ( t ) ‖ 0 < ∞

(30) ⇒ ∂ t 2 u , ∇ ∂ t u , ∂ t θ ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) .

Repeating the computation in (26), we have

(31) a ∫ 0 t ∫ Ω | ∇ ∂ t θ | 2 d x d τ ≤ ‖ ( ∂ t u , ∂ t θ ) ( 0 ) ‖ 0 2 + L ∫ 0 t ∫ Ω | ∂ t 2 u | 2 + | ∂ t u | 2 + | ∂ t θ | 2 d x d τ ≤ ‖ ( ∂ t u , ∂ t θ ) ( 0 ) ‖ 0 2 + L ( T + 1 ) ∫ 0 t ‖ ( ∂ t u , ∂ t θ ) ( τ ) ‖ 0 2 d τ < ∞

(32) ⇒ ∇ ∂ t θ ∈ L 2 ( [ 0 , T ] ; L 2 ( Ω ) ) .

Hence, (30), (32) imply ∂ t θ ∈ L 2 ( [ 0 , T ] ; H 0 1 ( Ω ) ) , this together with θ ∈ L 2 ( [ 0 , T ] ; H 0 1 ( Ω ) ) give that θ ∈ H 1 ( [ 0 , T ] ; H 0 1 ( Ω ) ) ↪ C ( [ 0 , T ] ; H 0 1 ( Ω ) ) . Since u solves

∂ t 2 u − Δ u + b ∇ θ = g ( u )

with a global Lipschitz g and g(0) = 0, we get the following elliptic equation for u:

(33) Δ u = ∂ t 2 u + b ∇ θ − g ( u ) ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) , u ( x , t ) = 0 , on ∂ Ω × ( 0 , + ∞ ) ,

therefore u ∈ C([0, T]; H ²(Ω)). Similarly, θ solves

∂ t θ − a Δ θ + b ∂ t div u = h ( θ ) ,

with a global Lipschitz h and h(0) = 0, hence θ solves the following elliptic equation

(34) a Δ θ = ∂ t θ + b ∂ t div u − h ( θ ) ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) , θ ( x , t ) = 0 , on ∂ Ω × ( 0 , + ∞ ) ,

and θ ∈ C([0, T]; H ²(Ω)). Collecting these results above, we have u , θ ∈ C ( [ 0 , T ] ; H 2 ( Ω ) ∩ H 0 1 ( Ω ) ) , ∂ t u ∈ C ( [ 0 , T ] ; H 0 1 ( Ω ) ) , ∂ t 2 u , ∂ t θ ∈ C ( [ 0 , T ] ; L 2 ( Ω ) ) . Therefore (u, θ) is a strong solution on Ω × [0, T]. Since T can be arbitrarily large, (u, θ) is a global strong solution for (1)– (4) with global Lipschitz nonlinear terms.

Step (ii) Global weak solutions for defocusing energy critical nonlinear terms

Based on the result in Step (i), we are able to handle with the original system (1)– (4). For g ( u ) = − u 1 5 , − u 2 5 , − u 3 5 and h satisfying (6), it is plausible that we choose a sequence of approximate functions: for k = 1, 2, 3, …, and define g ( k ) ( u ) = g 1 ( k ) ( u 1 ) , g 2 ( k ) ( u 2 ) , g 3 ( k ) ( u 3 ) and h ^(k)(θ) as follows

g j ( k ) ( u j ) = k 5 u j < − k , − u j 5 − k ≤ u j ≤ k , − k 5 u j > k , for j = 1,2,3 ,

and

h ( k ) ( θ ) = h ( − k ) θ < − k , h ( θ ) − k ≤ θ ≤ k , h ( k ) θ > k .

It is easy to check that, for all k = 1, 2, 3, …, g ^(k)(u), h ^(k)(θ) are both Lipschitz, and moreover, we have

G j ( k ) ( u j ) = − ∫ 0 u j g j ( k ) ( s ) d s ≥ 0 ,

and

h ( k ) ( θ ) θ ≤ C θ 2 .

Now we firstly assume that (u ₍₀₎, u ₍₁₎, θ ₍₀₎) in D(A), and consider the following approximate systems:

(35) ∂ t 2 u ( k ) − Δ u ( k ) + b ∇ θ ( k ) = g ( k ) ( u ( k ) ) in Ω × ( 0 , + ∞ ) , ∂ t θ ( k ) − a Δ θ ( k ) + b ∂ t div u ( k ) = h ( k ) ( θ ( k ) ) in Ω × ( 0 , + ∞ ) , u ( k ) ( x , t ) = 0 , θ ( k ) ( x , t ) = 0 on ∂ Ω × ( 0 , + ∞ ) , u ( k ) ( x , 0 ) = u ( 0 ) , ∂ t u ( k ) ( x , 0 ) = u ( 1 ) , θ ( k ) ( x , 0 ) = θ ( 0 ) , on Ω .

By the result in Step (i), for each k, we obtain a unique global strong solution (u ^(k), θ ^(k)) to (35). Let

E ( k ) ( u ( k ) ( t ) , θ ( k ) ( t ) ) = E 0 ( u ( k ) ( t ) , θ ( k ) ( t ) ) + ∑ j = 1 3 ∫ Ω G j ( k ) ( u j ( k ) ( t ) ) d x ,

then by the energy equality (18) and the condition (6), we have for any t > 0

E 0 ( u ( k ) ( t ) , θ ( k ) ( t ) ) ≤ E ( k ) ( u ( k ) ( t ) , θ ( k ) ( t ) ) ≤ E ( k ) ( u ( k ) ( 0 ) , θ ( k ) ( 0 ) ) + C ∫ 0 t ‖ θ ( k ) ( τ ) ‖ L 2 ( Ω ) 2 d τ ≤ E ( k ) ( u ( 0 ) , θ ( 0 ) ) + C ∫ 0 t E ( k ) ( u ( k ) ( τ ) , θ ( k ) ( τ ) ) d τ .

By Gronwall’s inequality with respect to E ^(k)(u ^(k)(t), θ ^(k)(t)), we get for any T > 0

sup 0 ≤ t ≤ T E 0 ( u ( k ) ( t ) , θ ( k ) ( t ) ) ≤ C

uniformly in k. Then by a procedure similar to (32), we are able to improve the regularity of θ ^(k) such that ∇θ ^(k) ∈ L ²(Ω × [0, T]) uniformly in k. Hence, (u ^(k), ∂ _t u ^(k), θ ^(k)) is bounded in H 0 1 ( Ω ) × L 2 ( Ω ) × H 0 1 ( Ω ) . Thus there exists ( u , ∂ t u , θ ) ∈ C [ 0 , T ] , H 0 1 ( Ω ) × C [ 0 , T ] , L 2 ( Ω ) × C [ 0 , T ] , H 0 1 ( Ω ) such that up to a subsequence

( u ( k ) , θ ( k ) ) → ( u , θ ) w e a k * in L ∞ [ 0 , T ] , H 0 1 ( Ω ) × H 0 1 ( Ω ) ,

∂ t u ( k ) ⇀ ∂ t u in L ∞ [ 0 , T ] , L 2 ( Ω ) .

Hence

(36) E 0 ( u ( t ) , θ ( t ) ) ≤ lim inf k → ∞ E 0 ( u ( k ) ( t ) , θ ( k ) ( t ) ) ,

uniformly for t ∈ [0, T]. Furthermore, by the Rellich’s theorem, we can get that

( u ( k ) , θ ( k ) ) → ( u , θ ) strongly in L 2 ( Ω × [ 0 , T ] ) ,

then

(37) ( u ( k ) , θ ( k ) ) → ( u , θ ) in D ′ ( Ω × ( 0 , T ) ) ,

(38) ( u ( k ) , θ ( k ) ) → ( u , θ ) a.e. ( x , t ) ∈ Ω × ( 0 , T ) .

For any φ = ( φ 1 , φ 2 , φ 3 ) ∈ C 0 ∞ ( Ω × ( 0 , T ) ) , we have

(39) ∫ 0 T ∫ Ω ∂ t u ∂ t φ − ∑ j = 1 3 ∇ x u j ∇ x φ j − b ∇ x θ φ d x d t = lim k → ∞ ∫ 0 T ∫ Ω ∂ t u ( k ) ∂ t φ − ∑ j = 1 3 ∇ x u j ( k ) ∇ x φ j − b ∇ x θ ( k ) φ d x d t = − lim k → ∞ ∫ 0 T ∫ Ω g ( k ) ( u ( k ) ) φ d x d t = − ∫ 0 T ∫ Ω g ( u ) φ d x d t ,

in the last line above, we have used Vitali’s convergence theorem. Similarly, for any ψ ∈ C 0 ∞ ( Ω × ( 0 , T ) ) , we have

(40) ∫ 0 T ∫ Ω ( θ ∂ t ψ − a ∇ x θ ∇ x ψ + b ∂ t u ∇ x ψ ) d x d t = − ∫ 0 T ∫ Ω h ( θ ) ψ d x d t .

Therefore by (36), (u, θ) is a global weak solution with finite energy. For the case with general initial data ( u ( 0 ) , u ( 1 ) , θ ( 0 ) ) ∈ H 0 1 ( Ω ) × L 2 ( Ω ) × H 0 1 ( Ω ) , we can approximate the initial data by data in D(A), then the global existence result in Theroem 1.1 can be derived by a standard approximation process. More specifically, we can find a sequence ( u ( 0 ) ( n ) , u ( 1 ) ( n ) , θ ( 0 ) ( n ) ) ⊆ D ( A ) such that

u ( 0 ) ( n ) → u ( 0 ) , in H 0 1 ( Ω ) ,

u ( 1 ) ( n ) → u ( 1 ) , in L 2 ( Ω ) ,

θ ( 0 ) ( n ) → θ ( 0 ) , in H 0 1 ( Ω ) .

For each n ≥ 1, taking ( u ( 0 ) ( n ) , u ( 1 ) ( n ) , θ ( 0 ) ( n ) ) as initial data, we can obtain solution (u ⁽ⁿ⁾, θ ⁽ⁿ⁾) to (1)– (4) with

u ( n ) ∈ C [ 0 , ∞ ) , H 0 1 ( Ω ) ∩ C 1 [ 0 , ∞ ) , L 2 ( Ω ) ,

θ ( n ) ∈ L 2 [ 0 , ∞ ) , H 0 1 ( Ω ) ∩ C [ 0 , ∞ ) , L 2 ( Ω ) .

Moreover, since ( u ( 0 ) ( n ) , u ( 1 ) ( n ) , θ ( 0 ) ( n ) ) is bounded in H 0 1 ( Ω ) × L 2 ( Ω ) × H 0 1 ( Ω ) , we have for any T > 0

(41) sup 0 ≤ t ≤ T E 0 ( u ( n ) ( t ) , θ ( n ) ( t ) ) + ‖ ∇ θ ( k ) ( t ) ‖ L 2 ( Ω ) ≤ C ,

uniformly in n. Thus there exists ( u , ∂ t u , θ ) ∈ C [ 0 , T ] , H 0 1 ( Ω ) × C [ 0 , T ] , L 2 ( Ω ) × C [ 0 , T ] , H 0 1 ( Ω ) such that up to a subsequence

( u ( n ) , θ ( n ) ) → ( u , θ ) w e a k * in L ∞ [ 0 , T ] , H 0 1 ( Ω ) × H 0 1 ( Ω ) ,

∂ t u ( n ) ⇀ ∂ t u in L ∞ [ 0 , T ] , L 2 ( Ω ) .

Hence

(42) E 0 ( u ( t ) , θ ( t ) ) ≤ lim inf n → ∞ E 0 ( u ( n ) ( t ) , θ ( n ) ( t ) ) ,

uniformly for t ∈ [0, T]. Based on (42), we can apply an analysis similar to (37)– (40) to get

∫ 0 T ∫ Ω ∂ t u ∂ t φ − ∑ j = 1 3 ∇ x u j ∇ x φ j − b ∇ x θ φ d x d t = − ∫ 0 T ∫ Ω g ( u ) φ d x d t ,

for any φ = ( φ 1 , φ 2 , φ 3 ) ∈ C 0 ∞ ( Ω × ( 0 , T ) ) , and

∫ 0 T ∫ Ω ( θ ∂ t ψ − a ∇ x θ ∇ x ψ + b ∂ t u ∇ x ψ ) d x d t = − ∫ 0 T ∫ Ω h ( θ ) ψ d x d t ,

for any ψ ∈ C 0 ∞ ( Ω × ( 0 , T ) ) . Therefore by (42), (u, θ) is a global weak solution with initial data in H 0 1 ( Ω ) × L 2 ( Ω ) × H 0 1 ( Ω ) . By the above steps, we complete the proof of Theorem 1.1.

3.2 Existence and uniqueness of local strong solutions

For the local well-posedness problem, one can expect higher regularity of solution. In order to state our result, we present the local Lipschitz condition in [18]:

Definition 3.1.

Suppose F is a nonlinear operator from a Banach space B into B. F is said to satisfy the local Lipschitz condition if for any positive constant M > 0, there is a positive constant L _M depending on M such that when u, v ∈ B, ‖u‖_B ≤ M and ‖v‖_B ≤ M

(43) ‖ F ( u ) − F ( v ) ‖ B ≤ L M ‖ u − v ‖ B .

Then we have the following local existence result for strong solution:

Proposition 3.2.

Suppose that the initial data u ( 0 ) ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) , u ( 1 ) ∈ H 0 1 ( Ω ) , θ ( 0 ) ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) , then there exists T > 0, such that the problem (1)– (4) adimits a unique strong solution (u, θ) satisfying

(44) u ∈ C [ 0 , T ) , H 2 ( Ω ) ∩ H 0 1 ( Ω ) ∩ C 1 [ 0 , T ) , H 0 1 ( Ω ) ,

and

(45) θ ∈ C [ 0 , T ) , H 2 ( Ω ) ∩ H 0 1 ( Ω ) ∩ C 1 [ 0 , T ) , L 2 ( Ω ) .

Proof.

We will apply Theorem 2.5.6 in [18] to prove Proposition 3.2. In order to do this, we need to show that the map

( u , ∂ t u , θ ) → ( g ( u ) , ∂ t ( g ( u ) ) , h ( θ ) )

is a nonlinear operator from D(A) to D(A), and satisfies the local Lipschitz condition. Since the estimates of u, ∂ _t u and θ are similar, we only show that

u → g ( u )

is a nonlinear operator from H 2 ( Ω ) ∩ H 0 1 ( Ω ) to H 2 ( Ω ) ∩ H 0 1 ( Ω ) , and satisfies the local Lipschitz condition.

Firstly note that for the space dimension n = 3, we have

(46) H 2 ( Ω ) ↪ C ( Ω ) , H 1 ( Ω ) ↪ L q ( Ω ) , for 2 < q ≤ 6 .

Since g ( u ) = − u 1 5 , − u 2 5 , − u 3 5 , it suffices to estimate u j 5 , j = 1, 2, 3. Then we compute by (46)

(47) ‖ u j 5 ‖ L 2 ( Ω ) ≤ u j L ∞ ( Ω ) 4 ‖ u j ‖ L 2 ( Ω ) ≲ u j H 2 ( Ω ) 5 ,

and

(48) ∂ x 2 u j 5 L 2 ( Ω ) ≤ C u j 4 ∂ x 2 u j L 2 ( Ω ) + u j 3 ( ∂ x u j ) 2 L 2 ( Ω ) ≤ C u j L ∞ ( Ω ) 4 ∂ x 2 u j L 2 ( Ω ) + u j L ∞ ( Ω ) 3 ∂ x u j L 4 ( Ω ) 2 ≤ C u j H 2 ( Ω ) 5 .

Combining (47) and (48) we have

g ( u ) H 2 ( Ω ) ≤ ∑ j = 1 3 u j 5 H 2 ( Ω ) ≤ C u H 2 ( Ω ) 5 .

Thus u → g(u) maps from H ²(Ω) to H ²(Ω). Furthermore, it is easy to see that suppg(u)(⋅, t) ⊆Ω if suppu(⋅, t) ⊆Ω, hence u → g(u) maps from H 2 ( Ω ) ∩ H 0 1 ( Ω ) to H 2 ( Ω ) ∩ H 0 1 ( Ω ) .

Next we turn to prove that u → g(u) is locally Lipschitz. Suppose that u , v ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) , and

‖ u ‖ H 2 ( Ω ) ≤ M , ‖ v ‖ H 2 ( Ω ) ≤ M ,

for fixed M > 0. Then we compute

(49) u j 5 − v j 5 L 2 ( Ω ) ≤ C u j L ∞ ( Ω ) + v j L ∞ ( Ω ) 4 ‖ u j − v j ‖ L 2 ( Ω ) ≤ C u j H 2 ( Ω ) + v j H 2 ( Ω ) 4 ‖ u j − v j ‖ H 2 ( Ω ) ≤ C M 4 ‖ u j − v j ‖ H 2 ( Ω ) ,

and

(50) ∂ x 2 u j 5 − v j 5 L 2 ( Ω ) ≤ C u j 4 ∂ x 2 u j − v j 4 ∂ x 2 v j L 2 ( Ω ) + u j 3 ∂ x u j 2 − v j 3 ∂ x v j 2 L 2 ( Ω ) ≤ C u j 4 ∂ x 2 u j − u j 4 ∂ x 2 v j L 2 ( Ω ) + u j 4 ∂ x 2 v j − v j 4 ∂ x 2 v j L 2 ( Ω ) + u j 3 ∂ x u j 2 − v j 3 ∂ x u j 2 L 2 ( Ω ) + v j 3 ∂ x u j 2 − v j 3 ∂ x v j 2 L 2 ( Ω ) ≤ C u j L ∞ ( Ω ) 4 ∂ x 2 u j − ∂ x 2 v j L 2 ( Ω ) + ∂ x 2 v j L 2 ( Ω ) u j 4 − v j 4 L ∞ ( Ω ) + ∂ x u j 2 L 2 ( Ω ) u j 3 − v j 3 L ∞ ( Ω ) + v j L ∞ ( Ω ) 3 ∂ x u j 2 − ∂ x v j 2 L 2 ( Ω ) ≤ C u j L ∞ ( Ω ) 4 ∂ x 2 u j − ∂ x 2 v j L 2 ( Ω ) + ∂ x 2 v j L 2 ( Ω ) u j L ∞ ( Ω ) + v j L ∞ ( Ω ) 3 u j − v j L ∞ ( Ω ) + ∂ x u j 2 L 2 ( Ω ) u j L ∞ ( Ω ) + v j L ∞ ( Ω ) 2 u j − v j L ∞ ( Ω ) + v j L ∞ ( Ω ) 3 ∂ x u j L ∞ ( Ω ) + ∂ x v j L ∞ ( Ω ) ∂ x u j − ∂ x v j L 2 ( Ω ) ≤ C M 4 ‖ u j − v j ‖ H 2 ( Ω ) .

(49) together with (50) yield

(51) ∂ x 2 u j 5 − v j 5 L 2 ( Ω ) ≲ M 4 ‖ u j − v j ‖ H 2 ( Ω ) .

Collecting the results in (49)– (51) we obtain that u → g(u) is locally Lipschitz in u , v ∈ H 2 ( Ω ) ∩ H 0 1 ( Ω ) . By similar analysis of ∂ _t u and θ we get that F maps from D(A) to D(A) and satisfies the locally Lipschitz condition. Hence all the conditions of Theorem 2.5.6 in [18] are satisfied and the result of Proposition 3.2 follows immediately. □

Corresponding author: Daoyin He, School of Mathematics, Southeast University, Nanjing, 211189, China, E-mail: 101012711@seu.edu.cn

Funding source: Jiangsu Provincial Scientific Research Center of Applied Mathematics

Award Identifier / Grant number: BK20233002

Acknowledgments

The author would like to thank Prof. H.-J. Gao and Prof. I. Witt for many helpful guidance and discussions.

Research ethics: Not applicable.
Informed consent: Not applicable.
Author contributions: All authors contributed significantly and equally to writing this article. All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.
Use of Large Language Models, AI and Machine Learning Tools: None declared.
Conflict of interest: Authors state no conflicts of interest.
Research funding: DH is supported by the Jiangsu Provincial Scientific Research Center of Applied Mathematics under Grant No. BK20233002, Southeast University Grant No. 2242023R40009. YS and KZ are supported by Jiangsu Provincial Department of Science and Technology under Grant No. BK20251063, SBK20250404551, Jiangsu Provincial Department of Education under Grant No. 23KJB110012, and Nanjing Institute of Technology under Grant No. YKJ202218.
Data availability: Data sharing does not apply to this article as no datasets were generated or analysed during the current study.

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Received: 2024-05-21

Accepted: 2025-09-18

Published Online: 2025-11-24

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https://doi.org/10.1515/math-2025-0205

Keywords for this article

3-D thermoelastic system; defocusing; energy critical; approximation; global weak solutions

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