Abstract
In the paper, we consider a system of thermoelasticity of type I with viscoelastic damping and nonlinear sources. By using the Galerkin method and the Banach fixed point theorem, we first prove the local existence and uniqueness of weak solution. Secondly, by extending the potential well method, we prove that the local solution exists globally if its initial position starts inside a family of “potential wells.” In particular, we also establish an explicit and optimal decay rate of energy driven by the decay rate of the relaxation function which includes exponential, algebraic, and logarithmic decay rates. Finally, by using the continuation theorem and the concavity arguments due to Levine (Trans Am Math Soc 192:1–21, 1974), we show that the local solution blows up at finite time in the sense of Ball (Q J Math Oxf 28(4): 473–486, 1977) if its initial position starts outside the “potential wells.” Further, an upper bound for the blow-up time is also given explicitly. Notice that our results imply a sharp result on the global existence and blow-up of the local weak solution and they also allow a relatively large class of relaxation functions that generalize the existing results in the literature.
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1 Introduction
Let \(\Omega \) be a bounded smooth domain of \(\mathbb {R}^n\) and consider the initial boundary value problem for the nonlinear system of thermoelasticity of type I with viscoelastic damping and nonlinear source term
where the displacement \({\textbf {u}}\) is a vector field and \(\theta \), the temperature, is the scalar function. The positive constants \(\mu ,\lambda \) are the Lamé coefficients satisfying \(\mu >0\), \(\lambda +2\mu >0\), and \(\alpha \), \(\beta \) are the coupling parameters. For simplicity, we shall assume that \(\alpha ,\beta \) are positive constants, but the other possible cases can be treated similarly.
Throughout the paper, we shall assume that:
-
The initial values:
$$\begin{aligned} \left( {\textbf {u}}_0,{\textbf {u}}_1,\theta \right) \in H_0^1(\Omega )^n\times L^2\left( \Omega \right) ^n\times L^2\left( \Omega \right) . \end{aligned}$$(1.2) -
The exponent \(p\) is a constant satisfying
$$\begin{aligned} 2<p\le \dfrac{2(n-1)}{n-2}\quad \text {if}\quad n\ge 3,\quad \text {and} \quad 2<p<\infty \quad \text {if}\quad n=1,2. \end{aligned}$$(1.3) -
The relaxation \(g:\mathbb {R}_+\rightarrow \mathbb {R}_+\) satisfies
- (G1):
-
\(g(0)>0\) and \(\mu -\int \limits _0^\infty g(s)\textrm{d}s=\ell >0\);
- (G2):
-
there exists a positive function \(H\in C^1\left( \mathbb {R}_+\right) \) and \(H\) is linear or strictly increasing and strictly convex \(C^2\) function on \(\left( 0,r\right] \), \(r\le g(0)\) with \(H(0)=H'(0)=0\) such that
$$\begin{aligned} g'(t)\le -\xi (t)H\left( g(t)\right) ,\quad \forall t\ge 0, \end{aligned}$$(1.4)where \(\xi \) is a positive nonincreasing differentiable function.
- (G3):
-
\(g\) satisfies
$$\begin{aligned} \int \limits _0^\infty g(\tau )\textrm{d}\tau<{\left\{ \begin{array}{ll} \displaystyle \frac{p(p-2)\mu -\alpha \beta }{(p-1)^2}& \text {if}\quad c_L\ge c_T,\\ \\ \displaystyle \frac{ p(p-2)\left( \lambda +2\mu \right) -\alpha \beta }{(p-1)^2}& \text {if}\quad c_L<c_T. \end{array}\right. } \end{aligned}$$(1.5)whence the exponent \(p\) satisfies
$$\begin{aligned} p>p_{\mu ,\lambda }={\left\{ \begin{array}{ll} 1+\sqrt{1+\frac{\alpha \beta }{\mu }}& \text {if}\quad c_L\ge c_T,\\ \\ 1+\sqrt{1+\frac{\alpha \beta }{\lambda +2\mu }}& \text {if}\quad c_L<c_T. \end{array}\right. } \end{aligned}$$(1.6)Here we denote by \(c_L=\sqrt{\lambda +2\mu }\) and \(c_T=\sqrt{\mu }\) the velocities of propagation of longitudinal and transversal waves.
In the absence of the viscoelastic damping and the source terms, the system (1.1) becomes the classical linear thermoelastic system due to Green and Naghdi [17]
From the mathematical point of view, the study of the asymptotic stability of solutions of the system of the form (1.7) was first studied by the pioneering work due to Dafermos [11], in which the author first proved the existence and uniqueness of the generalized solution and then showed that the energy of the generalized solution converges to zero if only if \(\Omega \) satisfies the following conditions:
-
(C)
There is no nontrivial eigenfunction \(\phi (x)\) of the Lamé system
$$\begin{aligned} {\left\{ \begin{array}{ll} -\mu \Delta u -\left( \lambda +\mu \right) \nabla \textrm{div}\,\phi = \gamma ^2\phi & \text {in}\quad \Omega ,\\ \textrm{div}\,\phi =0& \text {in}\quad \Omega ,\\ \phi =0& \text {on}\quad \partial \Omega . \end{array}\right. } \end{aligned}$$(1.8)
However no rate of decay was given. Also notice that the condition (1.8) generically holds true for smooth domain (see [3, 26, 47]), but fails when \(\Omega \) is a ball in \(\mathbb {R}^n\) (see [8, 26]). And it is now well-known that, in general, the total energy associated with the solution of (1.7) does not tend to zero as time goes to infinity.
Subsequently, there are a variety of results on the local and global well-posedness, as well as the uniform exponential decay rate of energy associated with solutions to some initial-boundary value problem for the linear thermoelastic system, both in one-dimension (see [9, 19, 23, 30, 41, 42] for example) and multi-dimension (see [11, 26, 29, 44]) and references therein.
The case of one-dimensional problem, the uniformly exponential decay rate of solutions to the linear thermoelastic system associated with various types of boundary conditions was studied extensively and completely thanks to the papers [9, 19, 22, 30, 41, 42] by some different methods. We refer here to the papers of Hansen [19] for the analysis of nonharmonic Fourier series, Kim [23] and Munoz Rivera [42] for the energy method and multiplier techniques, and Liu and Zheng [30] for the exponential stability of semigroups to establish the uniform exponential decay rate of the energy.
However, the situations are more complicated and much different in the higher-dimensional problem. There are some different approaches on the problem of uniform decay of energy of solution to linear thermoelastic system since the Dafermos’s pioneering work [11]. The case of the whole space \(\Omega =\mathbb {R}^3\), Dassios and Grillakis [14] studied the decay of energy for an isotropic linear thermoelasticity by decomposing the total energy associated with the longitudinal and thermal wave into three parts: kinetic energy, strain energy, and thermal energy and then showed that these three parts of the energy decay to zero as \(t\rightarrow \infty \) at the rate \(t^{-(m+3/2)}\) for sufficiently smooth data with compact support, where m is a suitable positive number depending on the initial data, while the transverse wave conserves its energy. Furthermore, Rivera [43] studied the decomposition of the displacement vector field in \(\mathbb {R}^n\) into two parts; one of them conserves its energy, the solenoidal part, or the nondissipative component, and the other that decays uniformly to zero as the rate \(t^{-n/2}\) as \(t\) goes to infinity, the dissipative component. In some special cases of domain, when it is radial symmetry like the spherical domain or the cylinder in \(\mathbb {R}^3\), Jiang et. al. [18] showed that the total energy decays exponentially.
The case of the bounded domain, Chirita [10] proved that the mean thermal energy tends to zero as time goes to infinity, but no rate of decay was obtained. The author also showed that the asymptotic equipartition occurs between the Cesàro means of the kinetic and strain energies. The method exploited in the paper is Levine’s refinement of the Lagrange-Brun identities. Another approach due to Rivera [44] is to decompose the displacement vector field into two parts: a curl-free part and the divergence-free part. The author showed that the energy decays exponentially when the initial data have a zero divergence-free part, whereas the total energy does not decay to zero uniformly if the divergence-free part of the initial data is not zero. In [26], by using the decoupling method, Lebeau and Zuazua proved a necessary and sufficient condition for the uniformly exponential decay rate of energy. Also from the paper, we know that the decay rate is not uniform under some certain geometric conditions on the domain \(\Omega \), that is, when \(\Omega \) is convex or such that there exists a ray of geometric optics in \(\Omega \) of arbitrarily large length that is always reflected perpendicularly on the boundary. Thus, for a general bounded domain, the uniform decay rate can occur if it has additional damping mechanisms. In this approach, we refer here to the paper of Pereira and Menzala [39, 40], and Mustafa [32] for the internal damping effective and the papers [4, 6, 33] for the memory damping type in the whole domain, Liu et. al. in [29, 31] for the linear and nonlinear damping on the boundary (see also [5]), and Oliveira and Charao [36] for weak localized dissipative term effective only in a neighborhood of part of the boundary.
Regarding the memory damping term, the study of the asymptotic stability of solutions of equations of linear viscoelasticity at large time was pioneered by Dafermos [12, 13], where the author showed that the generalized solution of the wave equation
tends to zero as time goes to infinity, where the memory kernel \(g\) is a non-negative, monotonic, nonincreasing, and convex function, but no rate of decay was given explicitly. Afterward, many efforts have been paid to the question of finding the uniform decay rate of energy and the ability to extend the memory kernel as generally as possible. To our best knowledge, the conditions (G1)–(G2) seem to be the most general conditions on \(g\) until now. We refer the reader to [1, 2, 25, 32, 34] for recent literature reviews. Recently, Mustafa [33] considered a thermoelastic system with viscoelastic damping, but without source of the form
where \(\mu ,\lambda >0\), and hence \(c_L=\sqrt{\lambda +2\mu }>c_T=\sqrt{\mu }\), that is, the velocity of propagation of longitudinal wave is greater than velocity of propagation of transversal wave, but it is still unknown for the case \(c_L\le c_T\), and \(g\) satisfies (G1) and
where \(H\) is a non-negative function, with \(H(0) = H'(0) = 0\), and \(H\) is strictly increasing and strictly convex on \((0, r]\), for some \(r > 0\). This is a special case of (1.4) when \(\xi (t)=1\). It is also well-known that PDEs with memory can be used to describe many physical phenomena when the effects of past values come into play, such as viscoelasticity in materials, population dynamics, or heat flow in real conductors, see [15, 35] and references therein for more detail.
Although there have been large literature results concerning the uniform decay rate of solution as well as its energy for the classical thermoelastic system, there are a few results on the blow-up of the solution of the thermoelastic system [20, 21, 24, 46], in which the authors used the concavity method of Levine [27] to show that the global solution fails to exist when the initial data are large enough, but the decay rate of the global solution was not studied well when the source term contributes to the system.
Motivated by these works, our aims in this manuscript are to not only study the stability but also the instability of the local weak solution of the classical thermoelasticity of type I with viscoelastic damping and nonlinear sources (1.1). More precisely, we first prove the existence and uniqueness of the local weak solution of (1.1) as well as its blow-up alternative. Our method was based on the Galerkin method and fixed-point arguments instead of using the Lions-Lax-Milgram theorem as in [11]. Next, by borrowing the idea of the potential well method due to Sattinger [45] (see also Payne and Sattinger [38]), we introduce the family of stable sets \(\mathscr {W}_\delta \) and unstable sets \(\mathscr {U}_\delta \) (see (2.2) and (2.3) below) and show that under some restricted conditions on the initial energy, the local solution exists globally when it starts from the “potential wells” \({\textbf {u}}_0\in \mathscr {W}_\delta \) and blows up in finite time if it begins outside the “potential wells,” that is, \({\textbf {u}}_0\in \mathscr {U}_\delta \). Further, we show in the former case an optimal decay rate of energy driven by the decay rate of the memory kernel \(g\) and give an upper bound for the blow-up time in the latter case. Also notice that by combining the alternative blow-up and the concavity arguments, we can conclude here the blow-up property of solution in the sense of Ball [7].
We end this section by the organization of the paper. Section 2 introduces some preparing lemmas and the family of stable and unstable sets. In Sect. 3, we give a local well-posedness result. Section 4 devotes to the global existence of weak solution and its optimal decay rates. Section 5 shows a sufficient condition for the blow-up in finite time and an upper bound for the blow-up time. Finally, we make some comments and open questions connected with our main results in Sect. 6.
2 Notations and preliminaries
2.1 Notations
Throughout the paper, we denote by \(\left\| \cdot \right\| _{p}\) the norm of the Lebesgue spaces \(L^p\left( \Omega \right) \) or \(L^p\left( \Omega \right) ^n:=\left( L^p\left( \Omega \right) \right) ^n\) for \(1<p<\infty \), and \(\left\| \cdot \right\| \) the norm of \(L^2\left( \Omega \right) \) or \(L^2\left( \Omega \right) ^n\), and \(\left\langle \cdot ,\cdot \right\rangle \) the inner product in \(L^2\left( \Omega \right) \) or \(L^2\left( \Omega \right) ^n\), and also the dual product between the Banach space \(X\) and its dual space \(X'\). Also denote by \(u'=u_t\) the time derivative of \(u=u(x,t)\).
2.2 Preparing lemmas
Lemma 2.1
(Jensen’s inequality) Let \(G: [a, b] \rightarrow \mathbb {R}\) be a convex function. Assume that the functions \(f: (0, L) \rightarrow [a, b]\) and \(h: (0, L) \rightarrow \mathbb {R}\) are integrable such that \(h(x) \ge 0\), for any \(x \in (0, L)\) and \(\int \limits _{0}^{L}h(x)\textrm{d}x=k>0\). Then
To prove the blow-up property, we shall use the concavity arguments due to [22, 28] which based on the following lemma.
Lemma 2.2
([22, 28]) Suppose that \(\phi (t)\in C^2\left[ 0,\infty \right) \) is a positive function satisfying the following inequality
where \(\epsilon \) is a positive constant. If \(\phi (0) > 0\), \(\phi '(0) > 0\), then \(\phi (t) \rightarrow \infty \) as \(t\rightarrow T_*\le T^*=\phi (0)/\epsilon \phi '(0)\).
2.3 Family of stable and unstable sets
For \(0<\delta \le \ell :=\mu -\int \limits _{0}^{\infty }g(\tau )\textrm{d}\tau \), we define the energy functionals \(J_\delta \) and \(I_\delta \) on \(H_0^1\left( \Omega \right) ^n\) by
It is known that the critical points of \(J\) are functions \(\phi \in H_0^1\left( \Omega \right) ^n\) satisfying the following Lamé system in a weak sense
Let \(\phi \in H_0^1\left( \Omega \right) ^n\backslash \{0\}\) and consider the fibering maps \(j:\mathbb {R}_+\rightarrow \mathbb {R}\) defined by \(j\left( z\right) =J(z\phi )\). Then we possess the following lemma.
Lemma 2.3
Assume that the Lamé coefficients \(\mu ,\lambda \) satisfy \(\mu >0\) and \(\lambda +2\mu >0\). If \(2< p<2^*:=\frac{2n}{n-2}\), then for any \(\phi \in H_0^1\left( \Omega \right) ^n\) and \(\phi \ne 0\), we possess:
-
(a)
\(\lim \limits _{z\rightarrow 0^+}j(z)=0\) and \(\lim \limits _{z\rightarrow \infty }j(z)=-\infty \);
-
(b)
there is a unique \(z_0=z_0(\phi )>0\) such that \(j'(z_0)=0\);
-
(c)
\(j''(z_0)<0\), that is, \(j\) attains its maximum value at \(z_0\). In addition, \(j\) increases on \((0,z_0)\) and decreases on \((z_0,\infty )\).
Proof
By definition of \(j\), we have
It is obvious that (a) holds since \(p>2\). On the other hand, we have
Then (b) and (c) follow by choosing \(z_0\) to be
The proof is complete. \(\square \)
We also define the Nehari manifolds and the potential depths (see [38]) as follows:
It follows from Lemma 2.3 that each half line starting from the origin of \(H_0^1(\Omega )\) intersects exactly once the manifold \(\mathscr {N}_\delta \) and hence it separates the two sets
Again by virtue of Lemma 2.3 and proceed similarly as in [38] we have
Notice that since \(p<2^*\), the embedding \(H_0^1\left( \Omega \right) ^n \hookrightarrow L^p\left( \Omega \right) ^n\) is compact and hence the infimum in (2.1) is attained (see [38] for example). So \(d_\delta \) is finite.
We also introduce the closed sublevels of \(J\) for level \(k\in \mathbb {R}\)
and define the family of the stable sets \(\mathscr {W}_\delta \) and unstable sets \(\mathscr {U}_\delta \) by
The following lemma will be useful in later arguments.
Lemma 2.4
Let \(p\) hold (1.3). Then we have
-
i)
\(\mathscr {W}_\delta \) is a bounded neighbourhood of \(0\) in \(H_0^1\left( \Omega \right) ^n\). More precisely, we have for each \(\phi \in \mathscr {W}_\delta \)
$$\begin{aligned} \left\| \nabla \phi \right\| ^2\le \frac{2p}{p-2}\cdot \frac{d_\delta }{\min \{\delta ,\lambda +\mu +\delta \}}. \end{aligned}$$(2.4) -
ii)
Let \(\phi \in \mathscr {U}_\delta \), then there exist a number \(z_0\in (0,1)\) such that \(z_0\phi \in \mathscr {N}_{\delta }\). In addition, we have
$$\begin{aligned} \left\| \nabla \phi \right\| ^2\ge \frac{2p}{p-2}\cdot \frac{d_\delta }{\max \{\delta ,\lambda +\mu +\delta \}}. \end{aligned}$$(2.5)As a consequence, we have \(0\notin \overline{\mathscr {U}_\delta }\).
Proof
For i). Let \(\phi \in \mathscr {W}_\delta \) then by definition we have \(I_\delta (\phi )>0\) and \(J_\delta (\phi )\le d_\delta \). Taking into account these facts and the estimate
we obtain (2.4).
For ii). Let \(\phi \in \mathscr {U}_\delta \) then by Lemma 2.3 there is a unique \(z_0=z_0(\phi )>0\) such that
On the other hand, since \(\phi \in \mathscr {N}_\delta ^-\), we have that \(I_\delta (\phi )<0\) which implies \(z_0\in (0,1)\). We next show that \(\phi \ne 0\). By the estimate
where \(S_p\) is the best Sobolev constant in the embedding \(H_0^1\left( \Omega \right) ^n\hookrightarrow L^p(\Omega )^n\),
It follows from \(I_\delta (\phi )<0\) that
Hence \(z_0\phi \in \mathscr {N}\). By (2.1) we have that
which implies (2.5). The proof is complete. \(\square \)
3 Local existence and uniqueness
We begin this section by the definition of weak solutions of (1.1).
Definition 3.1
(weak solution) A couple of functions \(({\textbf {u}},\theta )\) is a weak solution of system (1.1) on \([0,T]\) if
and satisfy for any \(\varphi \in H_0^1\left( \Omega \right) ^n\) and \(\psi \in H_0^1\left( \Omega \right) \)
almost every \(t\in [0,T]\) and the initial conditions \(u(0)=u_0\), \(u_t(0)=u_1\) and \(\theta (0)=\theta _0\).
The energy associated to system (1.1) is given by
where \(\left| \nabla {\textbf {u}}\right| ^2=\sum _{i=1}^n\left| \nabla {u}_i\right| ^2\) and
This energy decreases along the trajectory. More precisely
The local well-posedness and blow-up alternative are given by the following theorem.
Theorem 3.2
Assume that \(p\) holds (1.3), \(g\) holds (G1) and the initial data \(\left( {\textbf {u}}_0,{\textbf {u}}_1,\theta _0\right) \) satisfy (1.2). Then there exist a number \(T>0\) and a unique solution \(\left( {\textbf {u}},\theta \right) \) of (1.1) on \([0,T]\). In addition, if the maximal existence time
then the blow-up occurs at \(T_{\max }\), that is,
The proof of Theorem 3.2 is based on the Galerkin method and fixed-point arguments, but because we could not find any reference on its proof, so we present it here for self-contained purpose.
Let \(T>0\) and consider the Banach spaces \(\mathscr {H}_1= C\left( [0,T];H_0^1\left( \Omega \right) ^n\right) \cap C^1\left( [0,T];L^2\left( \Omega \right) ^n\right) \) and \(\mathscr {H}_2=C\left( [0,T];L^2\left( \Omega \right) \right) \cap L^2\left( 0,T;H_0^1\left( \Omega \right) \right) \) endowed with the norms
We begin with the following lemma.
Lemma 3.3
For every \(T>0\) and \(\left( {\textbf {u}},\theta \right) \in \mathscr {H}_1\times \mathscr {H}_2\) with initial data \(\left( {\textbf {u}}_0,{\textbf {u}}_1,\theta _0\right) \in H_0^1\left( \Omega \right) ^n\times L^2\left( \Omega \right) ^n\times L^2\left( \Omega \right) \), then there exists a unique pair of global weak solution \(\left( {\textbf {v}},\eta \right) \in \mathscr {H}_1\times \mathscr {H}_2\) satisfies
and
Proof
Uniqueness. The proof is proven by the energy method. Let \(\left( {\textbf {v}}_1,\eta _1\right) \) and \(\left( {\textbf {v}}_2,\eta _2\right) \) be two pairs of solutions of (3.3) which have the same initial data \(\left( {\textbf {u}}_0,{\textbf {u}}_1,\theta _0\right) \) in \(H_0^1\left( \Omega \right) ^n\times L^2\left( \Omega \right) ^n\times L^2\left( \Omega \right) \). Let \({\textbf {v}}={\textbf {v}}_1-{\textbf {v}}_2\) and \(\eta =\eta _1-\eta _2\), then \(\left( {\textbf {v}},\eta \right) \) solves
Multiplying the first and second equations of (3.3) with respect to \({\textbf {v}}_t\) and \(\eta \), respectively, and integrating by parts give
which implies \(\left( {\textbf {v}},\eta \right) =\left( {\textbf {0}},0\right) \).
Existence. Let \(\left\{ \varphi _i\right\} \) and \(\left\{ \psi _j\right\} \) be the Galerkin bases of the Hilbert spaces \(H_0^1\left( \Omega \right) ^n\) and \(H_0^1\left( \Omega \right) \), respectively. For \(k\ge 1\), let
and set
so that \({\textbf {v}}^k_0\rightarrow {\textbf {u}}_0\) in \(H_0^1\left( \Omega \right) ^n\) and \({\textbf {v}}^k_1\rightarrow {\textbf {u}}_1\) in \(L^2\left( \Omega \right) ^n\), and \(\eta ^k_0\rightarrow \theta _0\) in \(L^2\left( \Omega \right) \) as \(k\rightarrow \infty \).
For \(k\ge 1\) we find the approximating Galerkin solution \(\left( {\textbf {v}}^k,\eta ^k\right) \in V_k\times W_k\) of (1.1) of the forms
which satisfies \({\textbf {v}}^k(0)={\textbf {v}}_0^k\), \({\textbf {v}}_t^k(0)={\textbf {v}}_1^k\) and \(\eta ^k(0)=\eta ^k_0\), and holds
By the standard theory of ordinary differential equation, we obtain the existence and uniqueness of global solution \(\left( \nu _i^k,\gamma _j^k\right) \in C^2[0,T]\times C^1[0,T]\) of (3.5)–(3.6), which, in turn, it gives a unique \(\left( {\textbf {v}}^k,\eta ^k\right) \in C^2\left( [0,T];H_0^1\left( \Omega \right) ^n\right) \times C^1\left( [0,T];H_0^1\left( \Omega \right) \right) \) of the form (3.4).
Multiplying (3.5)–(3.6) by \(\left( \nu _{i}^k\right) '(t)\) and \(\gamma _j^k(t)\), respectively, and integrating over \([0,t]\subset [0,T]\), we obtain
where by using the Cauchy-Schwarz, the Sobolev and the Young inequalities and the fact that \(p\le {2(n-1)}/{(n-2)}\) and \({\textbf {u}}\in C\left( [0,T];H_0^1\left( \Omega \right) ^n\right) \) we find that
By using the identity
we have that
By virtue of the Cauchy-Schwarz and the Young inequalities, we obtain
Integrating by parts and then using the Cauchy-Schwarz inequalities, we obtain
Since \(g\) is non-negative and nonincreasing function, it follows from (3.7)–(3.12) that
Since \(\mu -\int \limits _{0}^{t}g(s)\textrm{d}s>0\), and \(\{{\textbf {v}}_0^k\}\), \(\{{\textbf {v}}_1^k\}\) and \(\{\eta _0^k\}\) are convergent sequences, by virtue of the Gronwall’s lemma, we obtain
for all \(k\ge 1\), where \(C_T>0\) is independent of \(k\). In turn, this uniform estimate implies that there exist the functions \({\textbf {v}},\eta \) and the subsequences of \(\left( {\textbf {v}}^k,\eta ^k\right) \) which still denoted by itself such that
Passing to the limit in (3.5)–(3.6) we obtain a pair of weak solution \(\left( {\textbf {v}},\eta \right) \) with the above regularity of the system
In addition, we next introduce the space as in Lions [28]
and consider the functions \(\varphi =\sum _{i=1}^{n}f_i\otimes \varphi _i\) and \(\psi =\sum _{j=1}^{m}f_j\otimes \psi _j.\) For \(k>m,n\), we have that
Let \(k\rightarrow \infty \) and notice that \(\{\varphi _i\}\) and \(\{\psi _j\}\) are dense in \(H_0^1\left( \Omega \right) ^n\) and \(H_0^1\left( \Omega \right) \), respectively, we obtain
for all \(\varphi \in L^2\left( 0,T;H_0^1\left( \Omega \right) ^n\right) \) and \(\varphi _t\in L^2\left( 0,T;L^2\left( \Omega \right) ^n\right) \) with \(\varphi (T)=0\), and \(\psi \in L^2\left( 0,T;H_0^1\left( \Omega \right) \right) \) and \(\psi _t\in L^2\left( 0,T;L^2\left( \Omega \right) \right) \) with \(\psi (T)=0\).
Initial data. We next show that \(\left( {\textbf {v}},\eta \right) \) satisfies the initial data \({\textbf {v}}(0)=u_0\), \({\textbf {v}}_t(0)={\textbf {u}}_1\) and \(\theta (0)=\theta _0\). Applying the Aubin-Lions lemma, it follows from (3.13)–(3.14) that
which implies \({\textbf {v}}^k(0)\rightarrow {\textbf {v}}(0)\) strongly in \(L^2\left( \Omega \right) ^n\) and hence \({\textbf {v}}(0)={\textbf {u}}_0\).
From (3.14) and (3.15), we have that \(\left\{ \left\langle {\textbf {v}}_t^k,\varphi _i\right\rangle \right\} \) and \(\left\{ \left\langle \eta ^k,\psi _j\right\rangle \right\} \) are bounded in \(L^\infty \left( 0,T\right) \). In addition, we also deduce from
that \(\left\{ \frac{\textrm{d}}{\textrm{d}t}\left\langle {\textbf {v}}_t^k,\varphi _i\right\rangle \right\} \) and \(\left\{ \frac{\textrm{d}}{\textrm{d}t}\left\langle \eta ^k,\psi _j\right\rangle \right\} \) are bounded in \(L^\infty \left( 0,T\right) \). And hence, up to sub-sequences we have for any \(i,j=1,\ldots ,k\)
As a consequence, we have that for any \(\varphi \in H_0^1\left( \Omega \right) ^n\) and \(\psi \in H_0^1\left( \Omega \right) \)
So we have \({\textbf {v}}_t(0)={\textbf {u}}_1\) and \(\eta (0)=\theta _0\). Finally, we deduce from (3.17)–(3.18) that
Regularity. Applying the Aubin-Lions lemma, we have that
By [28, Lemma 8.1], we have
In addition, by using the cut-off functions and mollifiers techniques as in the proof of [28, Lemma 8.3], we obtain the energy identity
And then using the arguments as in the proof of [28, Theorem 8.2] we can get better regularity of \({\textbf {v}}\), that is
The proof is complete. \(\square \)
We now exploited the fixed-point arguments to complete the proof of Theorem 3.2.
Proof of Theorem 3.2
Let \(\left( {\textbf {u}}_0,{\textbf {u}}_1,\theta _0\right) \in H_0^1\left( \Omega \right) ^n\times L^2\left( \Omega \right) ^n\times L^2\left( \Omega \right) \), and let \(T>0\), we consider the closed set
where \(M\) is chosen later.
By Lemma 3.3, for each \(\left( {\textbf {u}},\theta \right) \in \mathscr {M}_T\), then we may define the operator \(\Phi \) on \(\mathscr {M}_T\) by
where \(\left( {\textbf {v}},\eta \right) \) is a unique solution of (3.3). We next show that \(\Phi :\mathscr {M}_T\rightarrow \mathscr {M}_T\) is a contractive map for suitable \(T\) and \(M\). Indeed, let \(\left( {\textbf {u}},\theta \right) \in \mathscr {M}_T\) and the corresponding solution \(\left( {\textbf {v}},\eta \right) =\Phi \left( {\textbf {u}},\theta \right) \) for \(t\in [0,T]\). The energy identities give
Proceed similarly as above, we obtain
Since \(g\) is nonnegative and decreasing function, by choosing \(T\) sufficiently small, we obtain
Choosing \(M^2=4\left( \left\| {\textbf {u}}_1\right\| ^2 + {\mu }\left\| \nabla {\textbf {u}}_0\right\| ^2+({\lambda +\mu })\left\| \textrm{div}\,{\textbf {u}}_0\right\| ^2+\left\| \theta _0\right\| ^2\right) \) and picking \(T\) small so that
then we obtain \(\left\| {\textbf {v}}\right\| _{\mathscr {H}_1}^2+\left\| \eta \right\| _{\mathscr {H}_2}^2\le M^2.\) And hence, \(\Phi :\mathscr {M}_T\rightarrow \mathscr {M}_T\) is well-defined.
We next show that \(\Phi \) is a contractive mapping. Let \(\left( {\textbf {u}}_1,\theta _1\right) \) and \(\left( {\textbf {u}}_2,\theta _2\right) \) in \(\mathscr {M}_T\) and \(\left( {\textbf {v}}_1,\eta _1\right) =\Phi \left( {\textbf {u}}_1,\theta _1\right) \) and \(\left( {\textbf {v}}_2,\eta _2\right) =\Phi \left( {\textbf {u}}_2,\theta _2\right) \) be corresponding solutions of (3.3). Setting \({\textbf {v}}={\textbf {v}}_1-{\textbf {v}}_2\) and \(\eta =\eta _1-\eta _2\), then we have that
for all \(\left( \varphi ,\psi \right) \times H_0^1\left( \Omega \right) ^n\times H_0^1\left( \Omega \right) \). Taking \(\left( \varphi ,\psi \right) =\left( {\textbf {v}}_t(t),\eta (t)\right) \) and noticing that
where \(\xi (t)\) is a nonnegative function obtained by the Lagrange theorem and
Since \(2<p<\frac{2(n-1)}{n-2}\) and \(\frac{p-2}{2(p-1)}+\frac{1}{2(p-1)}+\frac{1}{2}=1\), by virtue of the Hölder inequality we have
Using similar argument as in (3.9) we obtain
and arguing similarly as in the proof of Lemma 3.3, we arrive at
which, in turn, \(\Phi \) is contractive mapping for \(T\) sufficiently small. Thus by the Banach contraction mapping theorem, there exists a unique \(\left( {\textbf {u}},\theta \right) \in \mathscr {H}_1\times \mathscr {H}_2\) such that \(\left( {\textbf {u}},\theta \right) =\Phi \left( {\textbf {u}},\theta \right) \), that is, the system (1.1) admits a unique solution \(\left( {\textbf {u}},\theta \right) \) on \([0,T]\).
Blow-up alternative. The last statement of the theorem can be proven by standard continuation arguments as in [37]. Suppose by contradiction that \(T_{\max }<\infty \) and
then there is a sequence \(\{t_n\}\), \(t_n\rightarrow T_{\max }^-\) and a constant \(C\) such that
For each \(n\), there exists a unique solution of (1.1) with initial data \(\left( {\textbf {u}}(t_n),{\textbf {u}}_t(t_n),\theta (t_n)\right) \) on \(\left[ t_n,t_n+T^*\right] \), where \(T^*>0\) depending only on \(C\). And hence we must have \(T_{\max }<t_n+T^*\) for sufficiently large \(n\). This contradicts to the definition of \(T_{\max }\).
The proof of Theorem 3.2 is complete. \(\square \)
4 Global existence and uniform decay of energy functional
Our main result of this section is the following theorem which shows the global existence and its stability.
Theorem 4.1
Let \(p\) hold (1.3), \(g\) hold (G1), (G2), and the initial data \(\left( {\textbf {u}}_0, {\textbf {u}}_1,\theta _0\right) \in H_0^1\left( \Omega \right) ^n\times L^2\left( \Omega \right) ^n\times L^2\left( \Omega \right) \) with
Then we have \(T_{\max }=\infty \). In addition, if we impose more restriction on \(E(0)\), that is,
then \(E(t)\) decays generally and uniformly. More precisely, there exist positive constants \(k_1,k_2\) to be such that
Remark 4.2
The estimate (4.2) shows a general decay of the total energy where its decay rates are driven by the decay rate of relaxation function \(g\). See the Sect. 6 for more comments on this rates.
The proof of Theorem 4.1 will be divided into two subsections where the first one is devoted to the global existence and the second one presents the decay rates of global solution.
4.1 Global existence
Let \({\textbf {u}}_0\in \mathscr {W}_\delta \), by contradiction arguments as in [38], it is not difficult to show that \({\textbf {u}}(t)\) lies in \(\mathscr {W}_\delta \) for every \(t\in [0,T_{\max })\). To prove that \(T_{\max }=\infty \), by the continuation principle, it suffices to show that
Indeed, from the energy identity we have
where \(E(t)\) is given by (3.1). Integrating it over \([0,t]\), we have
where \(D(t)=-\frac{1}{2}\left( g'\circ \nabla {\textbf {u}}\right) (t) + \frac{1}{2}g(t)\left\| \nabla {\textbf {u}}(t)\right\| ^2 + \frac{\alpha }{\beta }\left\| \nabla \theta (t)\right\| ^2\ge 0\). On the other hand, from the definition of \(E(t)\), we have
And, from the definition of \(J_\ell \) and \(I_\ell \), we have
for any \(0<\delta \le \ell \). Here we used the fact that \({\textbf {u}}(t)\in \mathscr {W}_\delta \).
Combining these facts, we arrive at
which, in turn, implies that \(T_{\max }=\infty \).
4.2 Uniform exponential decay of energy
We begin with the following lemma which plays important role in further development.
Lemma 4.3
With the assumptions of Theorem 4.1, we possess
and
Proof
It then follows from Lemma 2.3 that there is a number \(z_0>1\) such that \(z_0{\textbf {u}}(t)\in \mathscr {N}_\delta \). As a consequence, one has
which implies
By the variational characterization of potential depth \(d_\delta \) and again, by \(I_\delta (z_0 {\textbf {u}})=0\), we obtain
On the other hand, since \({\textbf {u}}_0\in \mathscr {W}_\delta \) and by the definition of \(I_\delta \) and \(J_\delta \) we also have
The last two estimates give
Combining (4.3) and (4.4) we arrive at
As a consequence, by the definition of \(I_\delta \) we have
The proof is complete. \(\square \)
We next define the auxiliary functions \(\Phi \) and \(\Psi \) by
The next two lemmas give the estimates for \(\Phi '(t)\) and \(\Psi '(t)\).
Lemma 4.4
(Estimate \(\Phi '\)) We possess
where \(h=\epsilon g-g'\) and \(C_\epsilon \) is positive and finite number thanks to
and \(C_\delta (\lambda ,\mu )\) is a positive constant given by
Proof
Multiplying (1.1) by the usual multiplier \({\textbf {u}}\), we have that
By using the identity
and the fact that \(\mu -\int _{0}^{\infty }g(s)\textrm{d}s=\ell \), we obtain
Applying the Fubini theorem and then using the Cauchy-Schwarz and the Young inequalities, the viscoelastic term yields for any \(\eta _1>0\)
where by setting \(h=\epsilon g-g'\)
And hence, it yields
Using the integration by parts and the fact that \(\left| \textrm{div}\,{\textbf {u}}\right| \le \left| \nabla {\textbf {u}}\right| \), the fourth term on the right side of (4.7) yields for any \(\eta _1>0\)
By Lemma 4.3, the nonlinearity term yields
Combining (4.8), (4.9), and (4.10), we deduce from (4.7) that
where \(C_\delta (\lambda ,\mu )\) given by
is a positive constant provided that
The proof follows by choosing \(\eta _1=C_\delta (\lambda ,\mu )/(1+\alpha )\). \(\square \)
Lemma 4.5
(Estimate \(\Psi '\)) For any \(\eta _2>0\), we have that
where \(h=\epsilon g-g'\), \(C_\epsilon \) as in Lemma 4.4, \(C_1(\lambda ,\mu )\) and \(C_2(\lambda ,\mu )\) are positive constants
Proof
Direct computations yield
We next estimate \(I_i\) for \(i=1,2,3\). Multiplying (1.1) by the viscoelastic multiplier \(\int _0^tg(t-s)\left( {\textbf {u}}(s)-{\textbf {u}}(t)\right) \textrm{d}s\) and integrating by parts give us
Proceed as in (4.8), we have for any \(\eta _2>0\)
Again, the use of Cauchy-Schwarz and the Young inequalities and the fact that \(\left| \textrm{div}\,{\textbf {u}}\right| \le \left| \nabla {\textbf {u}}\right| \) give
Integration by parts and again by \(\left| \textrm{div}\,{\textbf {u}}\right| \le \left| \nabla {\textbf {u}}\right| \) we have for any \(\eta _2>0\)
Since \(2<p<{2(n-1)}/{(n-2)}\), applying the Cauchy-Schwarz, the Poincaré and the Young inequalities we obtain
Here we used the fact that \({\textbf {u}}(t)\in \mathscr {W}_\delta \) and by Lemma 2.4
The last term in (4.12) yields
Here we applied the Cauchy-Schwarz and the Young inequalities for the last estimate.
Combining all these facts, (4.12) yields
For \(I_2\). Since \(g\) is nonincreasing relaxation function, using the Cauchy-Schwarz, the Young, and the Poincaré inequalities we obtain for any \(\eta _1>0\)
The proof follows from (4.11), (4.13), and (4.14) \(\square \)
The following lemma gives the sufficient conditions for the decay of energy.
Lemma 4.6
Assume that \(\int _{0}^{\infty }E(t)\textrm{d}t<\infty \) and \(L(t)\) is a functional which is equivalent to \(E(t)\) and holds
Then there exist positive constants \(k_1,k_2\) to be such that
where \(H_2(t)=\int _{t}^r\frac{1}{sH'(s)}\textrm{d}s\).
Proof
Since \(g\) and \(\xi \) are positive nonincreasing functions and \(H\) is a positive continuous function, then for fixed \(t_1>0\), there exist positive constants \(C_1,C_2\) such that
As a consequence, for all \(t\in [0,t_1]\)
thanks to \(g(t)\le g(0)\) for all \(t\ge 0\). This implies that for all \(t\ge t_1\)
due to the energy identity (3.2). It follows from (4.15) that
which implies for \(t\ge t_1\)
where \(F(t)=L(t)+c_3E(t)\) which is clearly equivalent to \(E(t)\).
Since \(\int _{0}^{\infty }E(s)\textrm{d}s<\infty \), we have that
By virtue of the Jensen inequality, we have that
Since \(H(0)=0\) and \(H\) is convex, by choosing \(\epsilon \) small such that \(\epsilon c(t)\le 1\), then we have
And hence, it deduce from (4.18) that
which implies
From (4.17) and (4.19), we arrive at
for \(\epsilon \) small enough and for all \(t\ge t_1\).
Multiplying both sides of (4.20) by \(G\left( \epsilon _1{E(t)}/{E(0)}\right) \), where \(0<\epsilon _1<r\), \(G\) is an increasing, convex function on \((0,r]\) and \(G(0)=0\), and define the product function
then it is not difficult to see that \(\mathscr {F}(t)\) is equivalent to \(E(t)\) and
due to \(E'(t)\le 0\), \(G'(t)>0\) and \(F(t)\sim E(t)>0\).
Applying the Young inequality, we find that
where \(H^*\) denotes the convex conjugate of \(H\) in the sense of Young, that is
Since \(H\) is convex, so \(H''>0\) or \(H'\) is an increasing function, we next choose \(G=H'\); then, it yields
It follows from (4.21) and (4.22) that
which implies
Define the auxiliary function
then (4.23) yields
Choosing \(\epsilon _1\) small enough then we obtain
where \(H_{1}(s)=sH'(s)\) is increasing function.
For \(0<\epsilon _2<\epsilon _1\), define the function
then we obtain
where \(\kappa _1>0\) depending on \(E(0)\). Integrating over \([t_1,t]\) yields
This implies that
where \(H_2(t)=\int _{t}^{r}\frac{\textrm{d}s}{H_1(s)}\) is a strictly decreasing function on \((0,r]\) and \(\lim \limits _{t\rightarrow 0^+}H_2(t)=\infty \) thanks to the properties of \(H\), and \(\epsilon _2\) was chosen to be such that \(\mathscr {H}(t_1)=\epsilon _2\mathscr {G}(t_1)/E(0)\le r\).
In addition, by taking \(t_1=g^{-1}(r)\) and choosing \(k\) to be such that \(\int _{t_1}^{2t_1}\xi (s)\textrm{d}s=k\int _{0}^{2t_1}\xi (s)\textrm{d}s\) then we find that
The proof follows from this fact and (4.24)–(4.25). \(\square \)
We now prove the second part of Theorem 4.1.
Proof of the second part of Theorem 4.1
Let us define the function
We shall show that the hypotheses of Lemma 4.6 hold. Firstly, by the definition of \(\Phi \) and \(\Psi \) (see (4.5) and (4.6)) it is not difficult to see that this functional is equivalent to \(E(t)\).
Secondly, we shall show that \(L(t)\) satisfies (4.15). It is noticed that since the relaxation \(g\) is positive and continuous, and \(g(0)>0\), for any \(t_0>0\) we have that
Taking this into account and by the energy identity (3.2) and Lemma 4.4 and Lemma 4.5 we have for any \(t\ge t_0\)
Since \(g'=\epsilon g-h\), we obtain
We first choose \(\eta _2={C_\delta \left( \lambda ,\mu \right) }/{4\Lambda _2C_1}>0\) and then take \(\Lambda _1\) large enough such that
and next choose \(\Lambda _2\) such that
Also notice that since \(\frac{\epsilon g^2(s)}{\epsilon g(s)-g'(s)}<g(s)\), by virtue of the Lebesgue dominated convergence theorem, we find that
And hence, we can pick a number \(0<\epsilon _0<1\) to be such that if \(\epsilon <\epsilon _0\), then
Finally we pick \(\epsilon =1/2\Lambda \) where \(\Lambda \) can be chosen large enough to be such that \(\epsilon <\epsilon _0\) and
then (4.26) yields
which implies (4.15).
We finally show that
For this purpose, let us introduce the functional
where \(f(t)=\int \limits _{t}^{\infty }g(s)\textrm{d}s\). It follows that
where by the virtue of Cauchy-Schwarz and Young inequalities, it yields for any \(\epsilon >0\)
On the other hand, since \(f(t)\) is a decreasing function, \(f(t)\le f(0)=\mu -\ell \). And hence, we arrive at
We next define the functional
then by (4.27) and (4.29) we obtain for all \(t\ge t_0\)
for some positive constant \(c\). This follows that
which implies (4.28). Thus the proof follows from the Lemma 4.6. \(\square \)
5 Finite time blow-up results
In this section, we deal with the instability of solutions to problem (1.1) in case the relaxation \(g:\mathbb {R}^+\rightarrow \mathbb {R}^+\) is nonincreasing differentiable function and satisfies
where \(\kappa _{\mu ,\lambda }\) is positive constant
The main result of this section is the following theorem.
Theorem 5.1
Let \(p\) hold (1.3) and (1.6), the relaxation function \(g\) satisfies (G1), (G3), and the initial data \(\left( {\textbf {u}}_0,{\textbf {u}}_1,\theta _0\right) \in H_0^1\left( \Omega \right) ^n\times L^2\left( \Omega \right) ^n\times L^2\left( \Omega \right) \) satisfy
where \(\ell _{\mu ,\lambda }=\kappa _{\mu ,\lambda }-\int _0^\infty g(\tau )\textrm{d}\tau >0\). Then the solution \(u(x,t)\) blows up at the finite time \(T_{\max }\), and an estimate for this blow-up time is given by
Proof
By contradiction, we assume that \(T_{\max }=\infty \) and define the function \(\phi :\left[ 0,T\right] \rightarrow \mathbb {R}^+\) by
for \(\eta >0\) and \(T_0>0\) chosen later. By direct computations, it yields
By the Cauchy-Schwarz and the Hölder inequalities, we have
Multiplying (1.1) by \({\textbf {u}}\) and integrating by parts, we obtain
Using the identity
and the following estimate due to the Hölder and the Young inequalities
it holds
By integrating by parts and noticing that \(\left| \textrm{div}\,{\textbf {u}}\right| \le \left| \nabla {\textbf {u}}\right| \), we have
It follows from (5.2)–(5.4) that
From (5.1) and (5.5), we deduce that
where
We shall show that \(\xi (t)\ge 0\) for \(t\in \left[ 0,T\right] \). Indeed, by the energy identity (3.1), we have
And hence, (5.7) yields
Choosing \(\varepsilon =\beta /p\) and noticing that \(\left| \textrm{div}\,{\textbf {u}}\right| \le \left| \nabla {\textbf {u}}\right| \), we obtain
where \(\kappa _{\mu ,\lambda }\) is a positive constant, due to (1.6), given by
On the other hand, since \({\textbf {u}}_0\in \mathscr {U}_\delta \), it is not difficult to prove that \({\textbf {u}}(t)\in \mathscr {U}_\delta \) and by Lemma 2.4
where \(E_\delta \) is given by
By choosing \(\eta =E_\delta -E(0)\) we deduce from (5.6) and (5.8) that
By Lemma 2.2, it remains to verify that \(\phi (0)>0\) and \(\phi '(0)>0\). It is obvious that \(\phi (0)>0\) for any \(T_0>0\). Next, by choosing \(T_0>0\) to be such that
and applying Lemma 2.2 with \(\epsilon =(p-2)/4\) then \(\phi (t)\rightarrow \infty \) as \(t\rightarrow T^*\), where \(T^*\) can be estimated by
More precisely, \(T_0\) can be any positive constant if \(\left\langle {\textbf {u}}_1,{\textbf {u}}_0\right\rangle > 0\) and
And hence, an upper bound for the blow-up time is
The proof is complete. \(\square \)
6 Conclusion and open questions
6.1 Conclusion
We have shown in Theorem 3.2 the local existence and uniqueness of a local solution \(\left( {\textbf {u}},\theta \right) \) of the problem (1.1) when the initial data \(\left( {\textbf {u}}_0,{\textbf {u}}_1,\theta _0\right) \in H_0^1\left( \Omega \right) ^n\times L^2\left( \Omega \right) ^n\times L^2\left( \Omega \right) \), the exponent \(p\) satisfies (1.3), and the relaxation function \(g\) satisfies
This condition shows that the local solution only exists when the relaxation is small, which is now the usual condition for the (local) well-posedness of the solution of PDEs involving the viscoelastic damping term. Further, because of the presence of both viscoelastic damping and the source term, it is also well-known that the global existence and blow-up of the local solution depend on the interaction between them. Roughly speaking, the local solution exists globally for the larger effects of viscoelastic damping and blows up in finite time for the larger effects of the source term.
In the second result, we have shown in Theorem 4.1 that if the initial data \(\left( {\textbf {u}}_0,{\textbf {u}}_1,\theta _0\right) \in H_0^1\left( \Omega \right) ^n\times L^2\left( \Omega \right) ^n\times L^2\left( \Omega \right) \) such that its initial position \({\textbf {u}}_0\) starts in the potential wells \(\mathscr {W}_\delta \) for \(0<\delta \le \ell \), then the local solution \(\left( {\textbf {u}},\theta \right) \) exists globally whenever the initial total energy \(E(0)\) is bounded. In addition, if we impose further conditions on the initial energy as in (4.1), that is,
then we get the optimal general decay for the total energy \(E(t)\)
It is first notice that by assumption of \(H\), in case \(\int _{0}^{\infty }\xi (t)\textrm{d}t=\infty \), then \(\lim \limits _{t\rightarrow \infty }E(t)=0\). In addition, decay rate of \(E(t)\) is driven by the decay rate of \(g\). Indeed, from (1.4), we have that
Define the function \({H}^*(t)=\int _{t}^r\frac{\textrm{d}s}{H(s)}\), then it is obvious that \({H}^*\) is strictly decreasing and convex on \(\left( 0,r\right] \) with \(\lim \limits _{t\rightarrow 0}{H}^*(t)=\infty \). And hence the last estimate yields
On the other hand, since \(H\) is strictly convex and \(H(0)=0\), we have \(0<H(t)-H(0)= H'(s)t\le H'(t)t\) for \(s\in [0,t]\) and hence
It shows that the decay rate of energy is optimal in the sense that it is consistent with the decay rate of \(g\). We also present here some special decay rates of \(g\) as in [34].
-
\(g\) is exponential decay, for example \(g(t) =c\exp ({-t^q})\) for \(0<q<1\), then \(g\) satisfies
$$\begin{aligned} g'(t)=\xi (t)H(g(t)), \end{aligned}$$where \(H(t)=t\) is linear function and \(\xi (t)=qt^{q-1}\) is a positive nonincreasing function for \(q<1\). And hence, the decay rate of energy \(E(t)\) is also exponential, that is,
$$\begin{aligned} E(t)\le k\exp \left( -k_1t^q\right) . \end{aligned}$$ -
\(g\) is algebraic decay, for example \(g(t)=c\left( t+1\right) ^{-q}\), for \(q>1\) satisfies (1.4) for \(c\) small and \(H(t)=t^m\), where \(1<m=({q+1})/{q}<2\) and \(\xi =1\). The decay rate of energy \(E(t)\) is given by
$$\begin{aligned} E(t)\le k\left( t+1\right) ^{-q}. \end{aligned}$$ -
\(g\) is logarithmic decay, for example \(g(t)=c(t+1)^{-1}\ln ^{-q}(t+1)\) for \(q>1\) satisfies (1.4) where \(H(t)=t^m\), \(1<m=({q+1})/{q}<2\) and \(\xi (t)=\left( t+1\right) ^\frac{1-q}{q}\ln \left( t+1\right) \). The decay rate of energy \(E(t)\) is given by
$$\begin{aligned} E(t)\le k\left( t+1\right) ^{-1}\ln ^{-q}(t+1). \end{aligned}$$
Our last result is Theorem 5.1, in which we have proved that when the initial data \(\left( {\textbf {u}}_0,{\textbf {u}}_1,\theta _0\right) \in H_0^1\left( \Omega \right) ^n\times L^2\left( \Omega \right) ^n\times L^2\left( \Omega \right) \) such that its initial position \({\textbf {u}}_0\) starts outside the potential wells \(\mathscr {W}_\delta \), that is, \({\textbf {u}}_0\in \mathscr {U}_\delta \) for \(0<\delta \le \ell \), then the local solution \(\left( {\textbf {u}},\theta \right) \) blows up in finite time under the condition
where \(\ell _{\mu ,\lambda }=\kappa _{\mu ,\lambda }-\int _{0}^{\infty }g(s)\textrm{d}s>0\) and \(\kappa _{\mu ,\lambda }<\mu \) is a positive constant given by
In addition, the blow-up time can be estimated as follows
This estimate implies the fact that the blow-up phenomena occur more quickly for higher values of \(p\).
Our results also derive a sharp result on the global existence and blow-up of the local solution of (1.1) in case \(p_{\mu ,\lambda }<p<2(n-1)/(n-2)\) and
Further in case \(E(0)<E_\delta =\min \{E_\delta ^1,E_\delta ^2\}\), we also obtain a sharp result on decay and blow-up.
6.2 Open questions
We state here a few questions and open problems related to our results:
-
1.
Whether the blow-up phenomena occur when \(2<p\le p_{\mu ,\lambda }\)?
-
2.
We do not know whether the energy decays when
$$\begin{aligned} \left( \frac{\min \{\ell ,\lambda +\mu +\ell \}}{\max \{\ell ,\lambda +\mu +\ell \}}\right) ^\frac{p}{p-2}{d_\ell }\le E(0)<d_\ell . \end{aligned}$$If it does, whether it decays uniformly? Notice that we have
$$\begin{aligned} \left( \frac{\min \{\ell ,\lambda +\mu +\ell \}}{\max \{\ell ,\lambda +\mu +\ell \}}\right) ^\frac{p}{p-2}{d_\ell }= d_\ell \quad \text {if}\quad c_L=c_T, \end{aligned}$$but
$$\begin{aligned} \left( \frac{\min \{\ell ,\lambda +\mu +\ell \}}{\max \{\ell ,\lambda +\mu +\ell \}}\right) ^\frac{p}{p-2}{d_\ell }< d_\ell \quad \text {if}\quad c_L\ne c_T. \end{aligned}$$ -
3.
In order to get the blow-up result, we need to impose the condition (1.5) on the relaxation function \(g\). This is a natural condition from the mechanical aspect; it tends to blow up when the effects of the damping are small. But we still do not know what happens if
$$\begin{aligned} \kappa _{\mu ,\lambda }\le \int \limits _{0}^{\infty }g(\tau )\textrm{d}\tau <\mu . \end{aligned}$$ -
4.
Finally, can we obtain a sharp result on the global existence and blow-up of local solution for the high initial energy \(E(0)\ge E_\delta \)? We could find such kind of result for the damped wave equation in Gazzola and Squassina [16].
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Acknowledgements
We would like to thank the anonymous referee for the careful reading of this paper and for the valuable suggestions to improve the presentation and the style of the paper. The first author is funded by Ho Chi Minh City University of Technology and Education (HCMUTE), Vietnam, under the Project No. T2023-74 entitled “Hyperbolic-parabolic systems: blow-up, global existence and uniform decay.” The second author is supported by Ho Chi Minh City University of Industry and Trade (HUIT), Vietnam. The third author is supported by University of Economics Ho Chi Minh City, Vietnam (UEH).
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Nhan, L.C., Nguyen, Y.V. & Truong, L.X. Optimal decay rate and blow-up of solution for a classical thermoelastic system with viscoelastic damping and nonlinear sources. Z. Angew. Math. Phys. 75, 175 (2024). https://doi.org/10.1007/s00033-024-02310-y
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DOI: https://doi.org/10.1007/s00033-024-02310-y