Research Article | Open Access
Zhiyong Ma, "Global Existence of the Higher-Dimensional Linear System of Thermoviscoelasticity", International Journal of Differential Equations, vol. 2011, Article ID 941679, 17 pages, 2011. https://doi.org/10.1155/2011/941679
Global Existence of the Higher-Dimensional Linear System of Thermoviscoelasticity
We obtain a global existence result for the higher-dimensional thermoviscoelastic equations. Using semigroup approach, we will establish the global existence of homogeneous, nonhomogeneous, linear, semilinear, and nonlinear, thermoviscoelastic systems.
In this paper, we consider global existence of the following thermoviscoelastic model: where the sign “” denotes the convolution product in time, which is defined by with the initial data and boundary condition The body is a bounded domain in with smooth boundary (say ) and is assumed to be linear, homogeneous, and isotropic. , and represent displacement vector and temperature derivations, respectively, from the natural state of the reference configuration at position and time . are Lamé's constants and the coupling parameters; denotes the relaxation function, is a specified “history,” and are initial data. denote the Laplace, gradient, and divergence operators in the space variables, respectively.
In what follows, we denote by the norm of , and we use the notation
When , system (1.1)–(1.4) is reduced to the thermoelastic system: In the one-dimensional space case, there are many works (see e.g., [4–8]) on the global existence and uniqueness. Liu and Zheng  succeeded in deriving in energy decay under the boundary condition (1.4) or or and Hansen  used the method of combining the Fourier series expansion with decoupling technique to solve the exponential stability under the following boundary condition: where is the stress. Zhang and Zuazua  studied the decay of energy for the problem of the linear thermoelastic system of type III by using the classical energy method and the spectral method, and they 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.
There are many works (see, e.g., [9, 12–15]) on exponential stability of energy and asymptotic stability of solution under different assumptions. The notation in this paper will be as follows. denote the usual (Sobolev) spaces on . In addition, denotes the norm in the space ; we also put . We denote by , the space of k-times continuously differentiable functions from into a Banach space , and likewise by , the corresponding Lebesgue spaces. denotes the Hölder space of -valued continuous functions with exponent in variable .
2. Main Results
Let the “history space” consist of -valued functions on for which Put with the energy norm where denotes the positive constant in , that is, Thus we consider the following thermoviscoelastic system: Let Since System (2.5) can be written as follows: We define a linear unbounded operator on by where and Set Then problem (2.8) can be formulated as an abstract Cauchy problem on the Hilbert space for an initial condition . The domain of A is given by where It is clear that is dense in .
Our hypotheses on can be stated as follows, which will be used in different theorems: (A1); (A2); (A3); (A4), and for any , .
We are now in a position to state our main theorems.
Theorem 2.1. Suppose that condition (A1) holds. Relaxation function satisfies (H1)–(H3). Then for any , there exists a unique global classical solution to system (2.8) satisfying .
Theorem 2.2. Suppose that condition (A2) holds. Relaxation function satisfies (H1)–( H3). Then for any , there exists a unique global classical solution to system (2.8) satisfying , that is,
Corollary 2.3. Suppose that condition (A3)or (A4) holds. Relaxation function satisfies (H1)–(H3). Then for any , there exists a unique global classical solution , to system (2.8).
Corollary 2.4. If and are Lipschitz continuous functions from into and , respectively, then for any , there exists a unique global classical solution , to system (2.8).
Theorem 2.5. Suppose relaxation function satisfies (H1)–(H3), , and , and satisfies the global Lipschitz condition on ; that is, there is a positive constant such that for all , Then for any , there exists a global mild solution to system (2.8) such that , that is,
Theorem 2.6. Suppose and , and is a nonlinear operator from into and satisfies the global Lipschitz condition on ; that is, there is a positive constant such that for all , Then for any , there exists a unique global classical solution , to system (2.8).
3. Some Lemmas
In this section in order to complete proofs of Theorems 2.1–2.6, we need first Lemmas 3.1–3.5. For the abstract initial value problem, where is a maximal accretive operator defined in a dense subset of a Banach space . We have the following.
Lemma 3.1. Let be a linear operator defined in a Hilbert space ,. Then the necessary and sufficient conditions for being maximal accretive are (i), (ii).
Proof. We first prove the necessity. is an accretive operator, so we have
Thus, for all ,
Letting , we get (i). Furthermore, (ii) immediately follows from the fact that is m-accretive.
We now prove the sufficiency. It follows from (i) that for all , Now it remains to prove that is densely defined. We use a contradiction argument. Suppose that it is not true. Then there is a nontrivial element belonging to orthogonal supplement of such that for all , It follows from (ii) that there is such that Taking the inner product of (3.5) with , we deduce that Taking the real part of (3.7), we deduce that , and by (3.6), , which is a contradiction. Thus the proof is complete.
Lemma 3.2. Suppose that is -accretive in a Banach space , and . Then problem (3.1) has a unique classical solution such that
Lemma 3.3. Suppose that , and Then problem (3.1) admits a unique global classical solution such that which can be expressed as
Proof. Since satisfies the homogeneous equation and nonhomogeneous initial condition, it suffices to verify that given by belongs to and satisfies the nonhomogeneous equation. Consider the following quotient of difference When , the terms in the last line of (3.13) have limits: It turns out that and the terms in the third line of (3.13) have limits too, which should be Thus the proof is complete.
Lemma 3.4. Suppose that , and Then problem (3.1) admits a unique global classical solution.
Lemma 3.5. Suppose that , and and for any , Then problem (3.1) admits a unique global classical solution.
Proof. We first prove that for any , the function given by the following integral:
belongs to . Indeed, we infer from the difference
that as ,
where we have used the strong continuity of and the absolute continuity of integral for .
Now it can be seen from the last line of (3.13) that for almost every exists, and it equals Thus, for almost every , Since and both belong to , it follows from (3.25) that for almost every , equals a function belonging to . Since is a closed operator, we conclude that and (3.25) holds for every . Thus the proof is complete.
To prove that the operator defined by (2.14) is dissipative, we need the following lemma.
Lemma 3.6. If the function is uniformly continuous and is in , then
Lemma 3.7. Suppose that the relaxation function satisfies and . If and , then
Proof. See, for example, the work by Liu in .
Lemma 3.8. Suppose relaxation function satisfies (H1)–(H3). The operator defined by (2.13) is dissipative; furthermore, , where is the resolvent of the operator .
Proof. By a straightforward calculation, it follows from Lemma 3.7 that
Thus, is dissipative.
To prove that , for any , consider that is, Inserting and obtained from (3.31), (3.33) into (3.34), we obtain By the standard theory for the linear elliptic equations, we have a unique satisfying (3.36).
We plug obtained from (3.31) into (3.35) to get Applying the standard theory for the linear elliptic equations again, we have a unique satisfying (3.37). Then plugging and just obtained from solving (3.36), (3.37), respectively, into (3.32) and applying the standard theory for the linear elliptic equations again yield the unique solvability of for (3.32) and such that . Thus the unique solvability of (3.30) follows. It is clear from the regularity theory for the linear elliptic equations that with being a positive constant independent of . Thus the proof is completed.
Lemma 3.9. The operator defined by (2.13) is closed.
Proof. To prove that is closed, let be such that Then we have By (3.40) and (3.44), we deduce By (3.42) and (3.46), we deduce By (3.47) and (3.49), we deduce and consequently, it follows from (3.41), that since is an isomorphism from onto . It therefore follows from (3.47) and (3.54) that By (3.43), (3.48), and (3.49), we deduce In addition, it follows from (3.39), (3.43), (3.51) that in the distribution. It therefore follows from (3.45) and (3.58) that and consequently, since is an isomorphism from onto . Thus, by (3.50), (3.52), (3.55), (3.57), (3.59), (3.60), we deduce Hence, is closed.
Lemma 3.10. Let be a linear operator with dense domain in a Hilbert space . If is dissipative and , the resolvent set of , then is the infinitesimal generator of a -semigroup of contractions on .
Lemma 3.11. Let be a densely defined linear operator on a Hilbert space . Then generates a -semigroup of contractions on if and only if is dissipative and .
Proof. See, for example, the work by Zheng in .
Proof of Theorem 2.1. By (2.2), it is clear that is a Hilbert space. By Lemmas 3.8–3.10, we can deduce that the operator is the infinitesimal generator of a -semigroup of contractions on Hilbert space . Applying the result and Lemma 3.2, we can obtain our result.
Proof of Theorem 2.2. we have known that the operator is the infinitesimal generator of a -semigroup of contractions on Hilbert space . Applying the result and Lemma 3.11, we can conclude that . If we choose operator , we can obtain and is dense in . Noting that by , we know that ; therefore, applying Lemma 3.1, we can conclude the operator is the maximal accretive operator. Then we can complete the proof of Theorem 2.2 in term of Lemma 3.3.
Proof of Corollary 2.4. We know that are Lipschitz continuous functions from into . Moreover, by (2.2), it is clear that is a reflexive Banach space. Therefore, . Hence applying Lemma 3.5, we may complete the proof of the corollary.
Proof of Theorem 2.5. By virtue of the proof of Theorem 2.2, we know that is the maximal accretive operator of a semigroup . On the other hand, satisfies the global Lipschitz condition on . Therefore, we use the contraction mapping theorem to prove the present theorem. Two key steps for using the contraction mapping theorem are to figure out a closed set of the considered Banach space and an auxiliary problem so that the nonlinear operator defined by the auxiliary problem maps from this closed set into itself and turns out to be a contraction. In the following we proceed along this line.
Let where is a positive constant such that . In , we introduce the following norm: Clearly, is a Banach space. We now show that the nonlinear operator defined by (4.1) maps into itself, and the mapping is a contraction. Indeed, for , we have where . Thus, that is, .
For , we have Therefore, by the contraction mapping theorem, the problem has a unique solution in . To show that the uniqueness also holds in , let be two solutions of the problem and let . Then By the Gronwall inequality, we immediately conclude that ; that is, the uniqueness in follows. Thus the proof is complete.
Proof of Theorem 2.6. Since is the maximal accretive operator, satisfies the global Lipschitz condition on . Let
Then is a Banach space, and is a densely defined operator from into . In what follows we prove that is -accretive in .
Indeed, for any , since is accretive in , we have that is, is accretive in . Furthermore, since is -accretive in , for any , there is a unique such that Now for any , (4.10) admits a unique solution . It turns out that Thus ; that is, is m-accretive in . Let be the semigroup generated by . If , then is unique classical solution of the problem. On the other hand, is also a classical solution in This implies that is a restriction of on . By virtue of the proof of Theorem 2.5, there exists a unique mild solution . Since is a restriction of on , and moreover, we infer from being an operator from to and Lemma 3.4 that is a classical solution to the problem. Thus the proof is complete.
This work was supported in part by Foundation of Shanghai Second Polytechnic University of China (no. A20XQD210006).
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