Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation

Abstract

This paper investigates the initial boundary value problem for a class of fourth order nonlinear damped wave equations modeling longitudinal motion of an elasto-plastic bar. By applying a suitable potential well-convexity method, we derive the global existence, asymptotic behavior and finite time blow up for the considered problem with more generalized nonlinear functions at subcritical initial energy level. Further for arbitrarily positive initial energy we give some sufficient conditions ensuring finite time blow up.

Keywords

Fourth order damped wave equation asymptotic behavior finite time blow up arbitrarily positive initial energy

1. Introduction

This paper focuses on the following initial boundary value problem of fourth order nonlinear damped wave equations $\begin{matrix} (1.1) & \begin{array}{l} u_{t t} + Δ^{2} u + γ u_{t} + \sum_{i = 1}^{n} \frac{\partial}{\partial x_{i}} σ_{i} (u_{x_{i}}) = 0, x \in Ω, t > 0, \\ u (x, 0) = u_{0} (x), u_{t} (x, 0) = u_{1} (x), x \in Ω, \\ u = \frac{\partial u}{\partial ν} = 0, x \in \partial Ω, t ⩾ 0, \end{array} \end{matrix}$ where $γ ⩾ 0$ , $Ω \subset R^{n}$ is a smooth bounded domain, ν is the outward unit normal vector on $\partial Ω$ , $\begin{matrix} (1.2) & u_{0} (x) \in H_{0}^{2} (Ω), u_{1} (x) \in L^{2} (Ω) \end{matrix}$ and the nonlinear function $σ (s) = (σ_{1} (s), \dots, σ_{n} (s))$ satisfies $\begin{matrix} (H) \{\begin{matrix} (i) & σ_{i} (s) \in C^{1} (R) and there exists a p > 1 such that \\ s (s σ_{i}^{'} (s) - p σ_{i} (s)) ⩾ 0, \forall s \in R, 1 ⩽ i ⩽ n; \\ (ii) & | σ_{i} (s) | ⩽ a | s |^{q}, a > 0, 1 < q < \frac{n + 2}{n - 2} for n ⩾ 3; 1 < q < \infty for n = 1, 2, 1 ⩽ i ⩽ n . \end{matrix} \end{matrix}$

1.1. Background

In the one-dimensional case, the original model equation of the considered problem (1.1) with $γ = 0$ of the form $\begin{matrix} (1.3) & u_{t t} + u_{x x x x} = a {(u_{x}^{2})}_{x} + f (x), a < 0 \end{matrix}$ came from the study of a weakly nonlinear analysis of elasto-plastic microstructure models, which can be used to describe the longitudinal motion of an elasto-plastic bar [1,2]. In the two-dimensional case, for some special nonlinear functions $σ (s)$ , problem (1.1) with $γ > 0$ is also related to a class of Kirchhoff-Boussinesq models considered by Chueshov and Lasiecka [6,7,9] to describe the dynamics of a plate with transverse shear effects, which naturally is a special case of a limit of the Mindlin–Timoshenko plates as the shear modulus tends to infinity [8]. Meanwhile, the existence of global attractors was obtained when the weak damping term $u_{t}$ of problem (1.1) was replaced by a strong damping term $u_{x x t}$ for one dimensional case [19]. In fact, there are numerous interesting literatures about the global attractors for this Kirchhoff–Boussinesq model equation with strong damping term, we refer reader to [27,28] and the references therein. Besides that, a sharp condition for global existence and nonexistence of solutions to problem (1.1) with nonlinear source term for $γ = 0$ at subcritical initial energy level was obtained in [10], which later was generalized by considering a class of more general nonlinearities in [15,22]. For the weak damping case $γ > 0$ , Yang [26] first introduced problem (1.1) and employed the potential well method to prove the global existence and asymptotic behavior of solutions for the case $0 < E (0) < d_{0} : = \frac{m - 1}{4 (m + 1)} {(\frac{1}{C_{*} b})}^{\frac{2}{m - 1}}$ under the assumption $| σ_{i} (s) | ⩽ b | s |^{m}$ . And the finite time blow up for $E (0) < 0$ was also shown in [26]. Subsequently, the well-posedness of problem (1.1) was further studied in [16] where the authors obtained the global existence and finite time blow up of solutions for the subcritical and critical initial energy $E (0) ⩽ d$ with the mountain pass level $d ⩾ d_{0}$ . Additionally, a numerical result for problem (1.1) was exhibited in [21] and the exponential stability of a finite-dimensional attractor and the existence of global attractor for problem (1.1) in one dimensional case were also derived in [17,19]. Most recently, a more generalized strain wave equation with strong damping term, nonlinear weak damping term and nonlinear external source term containing the strain wave equations mentioned above was dealt with in [13]. In which, the authors constructed the local existence, global existence, asymptotic behavior and finite time blow up of the solution to the initial boundary value problem for the following strain wave equation $\begin{matrix} (1.4) & u_{t t} - Δ u + Δ^{2} u + \sum_{i = 1}^{n} \frac{\partial}{\partial x_{i}} σ_{i} (u_{x_{i}}) - Δ u_{t} + | u_{t} |^{r - 1} u_{t} = f (u) \end{matrix}$ at subcritical initial energy, critical initial energy and supercritical initial energy levels in the frame work of variational arguments, which as a technique classifying the initial data recently has been used to treat various partial differential equations (see [3–5,14,18,20,24,25,29] and the references therein for instance).

1.2. Motivation

This paper still continues to focus on the global existence, asymptotic behavior and finite time blow up for problem (1.1) in the framework of variational arguments but with some generalized strain functions $σ (s) = (σ_{1} (s), \dots, σ_{n} (s))$ compared to the work conducted in [13,16,26]. In view of the assumption (H) on $σ (s)$ , we try to reveal the mechanism of the influence of the strain term on the dynamical behavior of the solution. In fact, the qualitative nature of solutions for the nonlinear wave equation not only depend on the initial conditions but also are related to the structure of the considered equation, especially whose damping term promotes the existence of global solution and also the asymptotic behavior of such global solution, while the nonlinear external term is conducive to the nonexistence of global solution (see [23] and the reference therein for more information). So, in this paper we shall consider the effect of the stain term $σ (s)$ as an external source term fulfilling some more weaker restrictive conditions (H) only associated with the weak damping term as did in [16,26] and try to extend some results in [13,16,26]. In order to make the motivation of the present paper clear, the coming contents report how the restrictions on the nonlinear terms $σ (s)$ are relaxed in our work.

Just for the simplicity to draw the differences among the conditions on the nonlinear term $σ (s)$ given in [13,16], we herein again show the assumption (H) on $σ (s)$ of the present paper as follows $\begin{matrix} (H) \{\begin{matrix} (i) & σ_{i} (s) \in C^{1} (R) and there exists a p > 1 such that \\ s (s σ_{i}^{'} (s) - p σ_{i} (s)) ⩾ 0, \forall s \in R, 1 ⩽ i ⩽ n; \\ (ii) & | σ_{i} (s) | ⩽ a | s |^{q}, a > 0, 1 < q < \frac{n + 2}{n - 2} for n ⩾ 3; \\ 1 < q < \infty for n = 1, 2, 1 ⩽ i ⩽ n \end{matrix} \end{matrix}$ and recall the assumption (H₁) required in [16] $\begin{matrix} (H_{1}) \{\begin{matrix} (i) & σ_{i} (s) \in C (R) \cap C^{1} (R^{+}), σ_{i} (s) > 0 for s \neq 0 and s σ_{i}^{'} (s) - σ_{i} (s) > 0 for s > 0, \\ (ii) & | σ_{i} (s) | ⩽ a | s |^{q}, a > 0, 1 < q < \frac{n + 2}{n - 2} for n ⩾ 3; \\ 1 < q < \infty for n = 1, 2; \\ (iii) & (p + 1) G_{i} (s) ⩽ s σ_{i} (s) for q ⩾ p > 1, \forall s \in R, G_{i} (s) = \int_{0}^{s} σ_{i} (τ) d τ \end{matrix} \end{matrix}$ and also the following assumption (H₂) required in [13] $\begin{matrix} (H_{2}) \{\begin{matrix} (i) & σ_{i} (s) \in C^{1} (R), σ_{i} (0) = σ_{i}^{'} (0) = 0, \\ (ii) & σ_{i} (s) are monotone, and convex for s > 0, concave for s < 0, \\ (iii) & | σ_{i} (s) | ⩽ a | s |^{q}, a > 0, 1 < q < \frac{n}{n - 2} for n ⩾ 3; \\ 1 < q < \infty for n = 1, 2; \\ (iv) & (q + 1) G_{i} (s) ⩽ s σ_{i} (s) for q > 1, \forall s \in R, for \forall s \in R, G_{i} (s) = \int_{0}^{s} σ_{i} (τ) d τ . \end{matrix} \end{matrix}$ Now, we describe the main differences among the assumptions (H₁), (H₂) and (H). Indeed, it is obvious to find that both (i) and (iii) in (H₁) are replaced by the assumption (i) in (H) and “ $σ_{i} (s) > 0$ for $s \neq 0$ ” in (H₁) is not required any more in (H). Also, the assumption (i) in (H) takes place of (i), (ii), (iv) in (H₂) and dose not need the restriction “ $σ_{i} (s) are monotone$ ” of (H₂). In this regard, the above differences imply that the assumption (H) introduced in this paper includes much wider nonlinear functions than that dose in (H₁) required in [16] and in (H₂) restricted in [13]. For instance, all the following example functions $\begin{matrix} \{\begin{matrix} (i) & σ_{i} (s) = a_{i} | s |^{p - 1} s, \\ (ii) & σ_{i} (s) = \pm a_{i} | s |^{p}, \\ (iii) & σ_{i} (s) = a_{i} | s |^{p - 1} s \pm b_{i} | s |^{p} \end{matrix} \end{matrix}$ satisfy the assumption (H), but only $σ_{i} (s) = a_{i} | s |^{p}$ in (ii) of above satisfies the assumption (H₁), and $σ_{i} (s) = a_{i} | s |^{p - 1} s$ in (i) of above fulfills the assumption (H₂), where both $a_{i}$ and $b_{i}$ are positive constants. Further, we also note that the assumption (H) is satisfied by $σ (s) = \pm a | s |^{2}$ in Equation (1.3) and its generalization $σ (s) = \pm a | s |^{p}$ .

Next we turn to discuss some obtained results of our work under the assumption (H) and also the corresponding improvements comparing with the conclusions in [13,16,26]. The further interest of our considered problem (1.1) is to reveal the potential effect of both internal dissipative term and external nonlinearity involving the initial data on the decay rate of the asymptotic behavior and finite time blow up of solutions. Our first concern is the asymptotic analysis of the global solution, that is the exponential decay. Although in view of the proof of Theorem 4.4 in [13] recording an exponential decay result for Equation (1.4) with $r > 1$ and subcritical initial energy, one can infer that the global solution to problem (1.1) decays exponentially, and we also observe that the approach performed in [13] only holds for $(q + 1) G_{i} (s) = s σ_{i} (s)$ , where q is the index in $| σ_{i} (s) | ⩽ a | s |^{q}$ . In other words, for $(q + 1) G_{i} (s) \neq s σ_{i} (s)$ , it is unclear that whether the asymptotic behavior of the global solution still holds. Particularly, the internal dissipative terms including the strong damping term and linear or non-linear weak damping term is the key point leading to the asymptotic behavior, but the exponential decay rate still remains unknown. Hence, in order to make it clear, in this paper we still consider problem (1.1) posed in [16,26]. The difference compared to [13] is that the strong damping term $Δ u_{t}$ in Equation (1.4) vanishes and the weak damping term in Equation (1.4) is linear ( $r = 1$ ) but with more generalized strain terms satisfying the assumption (H). Meanwhile, we also show that the exponential decay given in the proof of Theorem 2.1 in [26] still holds for $E (0) ⩾ d_{0} : = \frac{m - 1}{4 (m + 1)} {(\frac{1}{C_{*} b})}^{\frac{2}{m - 1}}$ , which enlarges the range of the initial data assuring the exponential decay in some senses. So, in this paper, based on the multiplier method we not only generalize the asymptotic behavior displayed in [26] from the case $\begin{matrix} 0 < E (0) < d_{0} : = \frac{m - 1}{4 (m + 1)} {(\frac{1}{C_{*} b})}^{\frac{2}{m - 1}} \end{matrix}$ to $\begin{matrix} 0 < E (0) ⩽ d_{1} : = \frac{m - 1}{2 (m + 1)} {(\frac{1}{C_{*} b})}^{\frac{2}{m - 1}} = 2 d_{0}, \end{matrix}$ if we take $p = q = m$ and $a = b$ in (2.5), but also demonstrate that this asymptotic behavior still holds for some generalized strain terms under assumption (H), which allows $(q + 1) G_{i} (s) \neq s σ_{i} (s)$ . The consideration mentioned above in this paper further reveals how the interplay of internal dissipative term, external nonlinearity and the initial data involved affects the qualitative behavior especially asymptotic behavior of the solution. Our second concern is the finite time blow up of the solution. Here, we employ a suitable potential well-convexity method to derive a finite time blow up result of solutions with subcritical initial energy independent of the damping term coefficient γ, i.e., Theorem 3.3, which removes the restrictions on the coefficient of the damping term required in [16]. Besides that, we establish a finite time blow up result for the arbitrarily positive initial energy $E (0) > 0$ (or the supercritical initial energy $E (0) > d$ ) by introducing a new auxiliary function and utilizing an adapted concave inequality (see Theorem 4.1), which extends the corresponding conclusions given in [16,26] from the subcritical and critical initial energy $E (0) ⩽ d$ to the supercritical initial energy $E (0) > d$ and also discovers the effect of the coefficient of the weak damping term, γ, on the finite time blow up for the arbitrarily positive initial energy $E (0) > 0$ compared to [13].

1.3. Organization

This paper is organized as follows. In Sect. 2, we define some functionals and sets, and then give some lemmas to state their properties. Section 3 deals with the global well-posedness and the asymptotic behavior of the solution for the subcritical initial energy $E (0) < d$ . Section 4 handles the finite time blow up of solutions with arbitrarily positive initial energy $E (0) > 0$ .

2. Preliminaries

This section aims to prepare for the further discussions. Throughout the present paper, we also use the notations $‖ \cdot ‖_{p} = ‖ \cdot ‖_{L^{p} (Ω)}$ , $‖ \cdot ‖ = ‖ \cdot ‖_{2}$ and the inner product $(u, v) = \int_{Ω} u v d x$ . Besides, the following essential functionals $\begin{array}{l} (2.1) & E (t) = \frac{1}{2} ‖ u_{t} ‖^{2} + \frac{1}{2} ‖ Δ u ‖^{2} - \sum_{i = 1}^{n} \int_{Ω} G_{i} (u_{x_{i}}) d x = \frac{1}{2} ‖ u_{t} ‖^{2} + J (u), \\ (2.2) & J (u) = \frac{1}{2} ‖ Δ u ‖^{2} - \sum_{i = 1}^{n} \int_{Ω} G_{i} (u_{x_{i}}) d x, \\ (2.3) & I (u) = ‖ Δ u ‖^{2} - \sum_{i = 1}^{n} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x, \\ d = inf_{u \in N} J (u), \\ N = {u \in H_{0}^{2} (Ω) ∣ I (u) = 0, u \neq 0} \end{array}$ for problem (1.1) are also introduced. Clearly for $u \in H_{0}^{2} (Ω)$ all above definitions are well defined.

Lemma 2.1.
Let $σ (s)$ satisfy (H). Then there holds $\begin{matrix} (2.4) & (p + 1) G_{i} (s) ⩽ s σ_{i} (s), \forall s \in R, 1 ⩽ i ⩽ n . \end{matrix}$
Proof.
We consider the following two cases to finish this proof.
If $s > 0$ , then (i) in (H) implies $\begin{matrix} s σ_{i}^{'} (s) ⩾ p σ_{i} (s), s > 0 \end{matrix}$ and $\begin{matrix} s σ_{i} (s) - \int_{0}^{s} σ_{i} (τ) d τ = \int_{0}^{s} τ σ_{i}^{'} (τ) d τ ⩾ p \int_{0}^{s} σ_{i} (τ) d τ = p G_{i} (s), s > 0, \end{matrix}$ which gives $\begin{matrix} (p + 1) G_{i} (s) ⩽ s σ_{i} (s), s ⩾ 0, 1 ⩽ i ⩽ n . \end{matrix}$

If $s < 0$ , then (i) in (H) gives $\begin{matrix} s σ_{i}^{'} (s) ⩽ p σ_{i} (s), s < 0, \end{matrix}$ and $\begin{matrix} s σ_{i} (s) - \int_{0}^{s} σ_{i} (τ) d τ = \int_{0}^{s} τ σ_{i}^{'} (τ) d τ ⩾ p \int_{0}^{s} σ_{i} (τ) d τ = p G_{i} (s), s < 0, \end{matrix}$ which gives $\begin{matrix} (p + 1) G_{i} (s) ⩽ s σ_{i} (s), \forall s < 0, 1 ⩽ i ⩽ n . \end{matrix}$
The above two cases display (2.4). □

Note that in the proof of Lemma 2.5–Lemma 2.7 in [16] only the assumption (ii) in (H₀) was used. Hence we have the following lemma immediately without proving them again.
Lemma 2.2 ([16]).

Let $σ (s)$ satisfy (H) and $u \in H_{0}^{2} (Ω)$ . Then there holds

if $0 < ‖ Δ u ‖ < r_{0}$ , then $I (u) > 0$ ;

if $I (u) < 0$ , then $‖ Δ u ‖ > r_{0}$ ;

if $I (u) = 0$ and $u \neq 0$ , then $‖ Δ u ‖ ⩾ r_{0}$ ,

where

\begin{matrix} r_{0} = {(\frac{1}{a C_{*}^{p + 1}})}^{\frac{1}{q - 1}}, C_{*} = sup_{u \in H_{0}^{2} (Ω) / 0} \frac{‖ \nabla u ‖_{q + 1}^{q + 1}}{‖ Δ u ‖^{q + 1}} . \end{matrix}

Lemma 2.3 ([16]).

Let $σ (s)$ satisfy (H). Then $\begin{matrix} (2.5) & d ⩾ d_{1} = \frac{p - 1}{2 (p + 1)} {(\frac{1}{a C_{*}})}^{\frac{2}{q - 1}} . \end{matrix}$

Lemma 2.4.
Let $σ (s)$ satisfy (H) and $u \in H_{0}^{2} (Ω)$ . Assume that $I (u) < 0$ . Then there at least exists a $λ^{} \in (0, 1)$ such that* $I (λ^{} u) = 0$ .
Proof.
From $I (u) < 0$ it implies $\begin{matrix} \sum_{i = 1}^{n} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x > ‖ Δ u ‖^{2} > 0 . \end{matrix}$ Let $\begin{matrix} φ (λ) : = \frac{1}{λ} \sum_{i = 1}^{n} \int_{Ω} u_{x_{i}} σ_{i} (λ u_{x_{i}}) d x, λ > 0 . \end{matrix}$ Then $φ (λ)$ is continuous on $0 < λ < \infty$ and $φ (1) > ‖ Δ u ‖^{2}$ . On the other hand, form (ii) in (H) it follows $\begin{array}{l} | φ (λ) | & ⩽ \frac{1}{λ^{2}} \sum_{i = 1}^{n} \int_{Ω} λ u_{x_{i}} σ_{i} (λ u_{x_{i}}) d x \\ ⩽ \frac{1}{λ^{2}} \sum_{i = 1}^{n} \int_{Ω} a | λ u_{x_{i}} |^{q + 1} d x \\ = λ^{q - 1} a \sum_{i = 1}^{n} ‖ u_{x_{i}} ‖_{q + 1}^{q + 1} \\ = λ^{q - 1} a ‖ \nabla u ‖_{q + 1}^{q + 1}, \end{array}$ which implies $\begin{matrix} lim_{λ \to 0} φ (λ) = 0 . \end{matrix}$ Hence that exists at least a $λ^{} \in (0, 1)$ such that $I (λ^{} u) = 0$ . □
Lemma 2.5.
Let* $σ (s)$ satisfy (H) and $u \in H_{0}^{2} (Ω)$ . Assume that $I (u) < 0$ . Then $\begin{matrix} (2.6) & I (u) < (p + 1) (J (u) - d) . \end{matrix}$
Proof.
From $I (u) < 0$ and Lemma 2.4 it follows that there exits a $λ^{} \in (0, 1)$ such that $I (λ^{} u) = 0$ . Let $\begin{matrix} g (λ) : = (p + 1) J (λ u) - I (λ u), λ > 0 . \end{matrix}$ Then $\begin{matrix} g (λ) = \frac{p - 1}{2} λ^{2} ‖ Δ u ‖^{2} + \sum_{i = 1}^{n} \int_{Ω} (λ u_{x_{i}} σ_{i} (λ u_{x_{i}}) - (p + 1) G_{i} (λ u_{x_{i}})) d x \end{matrix}$ and $\begin{array}{l} g^{'} (λ) & = (p - 1) λ ‖ Δ u ‖^{2} + \frac{1}{λ} \sum_{i = 1}^{n} \int_{Ω} λ u_{x_{i}} (λ u_{x_{i}} σ_{i}^{'} (λ u_{x_{i}}) - p σ_{i} (λ u_{x_{i}})) d x \\ ⩾ (p - 1) λ ‖ Δ u ‖^{2} > 0, λ > 0 . \end{array}$ Hence $g (λ)$ is strictly increasing on $0 < λ < \infty$ , which gives $g (1) > g (λ^{})$ , i.e., $\begin{matrix} (p + 1) J (u) - I (u) > (p + 1) J (λ^{} u) - I (λ^{} u) = (p + 1) J (λ^{} u) ⩾ (p + 1) d . \end{matrix}$ Hence the proof of this lemma is complete. □

Now for problem (1.1) we define $\begin{array}{l} W : = {u \in H_{0}^{2} (Ω) ∣ I (u) > 0, J (u) < d} \cup {0} \end{array}$ and $\begin{array}{l} V : = {u \in H_{0}^{2} (Ω) ∣ I (u) < 0, J (u) < d} . \end{array}$ In addition, we define $\begin{array}{l} W^{'} : = {u \in H_{0}^{2} (Ω) ∣ I (u) > 0} \cup {0} \end{array}$ and $\begin{array}{l} V^{'} : = {u \in H_{0}^{2} (Ω) ∣ I (u) < 0} . \end{array}$

Throughout the present paper, we also employ the definition of weak solutions to problem (1.1) given in Definition 1.1 in [16], especially the energy inequality there $\begin{matrix} (2.7) & E (t) + γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ = E (0), t \in [0, T) . \end{matrix}$

Since the proofs of some theorems that we will show are very similar as those in [16], we only give the conclusions without proving them. Now we can show some invariant sets as follows. Theorem 2.1.
Let $σ (s)$ satisfy (H) and ( 1.2 ) hold. Then there holds
if $0 < E (0) < d$ and $u_{0} (x) \in W^{'}$ , then all weak solutions to problem ( 1.1 ) belong to $W^{'}$ , especially W;

if $0 < E (0) < d$ and $u_{0} (x) \in V^{'}$ , then all weak solutions to problem ( 1.1 ) belong to $V^{'}$ , especially V;

if $E (0) < 0$ or $E (0) = 0$ , $u_{0} \neq 0$ , then all weak solutions to problem ( 1.1 ) belong to V.

3. Problem (1.1) for subcritical initial energy $E (0) < d$

3.1. Global existence and asymptotic behavior for $E (0) < d$

In this subsection, we give a global existence theorem (Theorem 3.1) without proving it, as the proof can be derived through replacing (H₀) by (H) and Lemma 2.1. And we will focus on the asymptotic behavior, i.e., the exponential decay in this part.

Theorem 3.1.
Let $σ (s)$ satisfy (H) and ( 1.2 ) hold. Assume $u_{0} \in W$ , then problem ( 1.1 ) admits a global weak solution $u (t) \in C (0, \infty; H_{0}^{2} (Ω))$ with $u_{t} (t) \in L^{\infty} (0, \infty; L^{2} (Ω))$ and $u \in W$ for $0 ⩽ t < + \infty$ .

As did in Theorem 4.1 in [16], let ${w_{j} (x)}$ be a system of base functions in $H_{0}^{2} (Ω)$ . Then the approximate solutions to problem (1.1) can be constructed as follows $\begin{matrix} u_{m} (x, t) = \sum_{j = 1}^{m} g_{j m} (t) w_{j} (x), m = 1, 2, \dots \end{matrix}$ satisfying $\begin{array}{l} (3.1) & (u_{m t t}, w_{s}) + γ (u_{m t}, w_{s}) + a (u_{m}, w_{s}) - b (u_{m}, w_{s}) = 0, \\ (3.2) & u_{m} (x, 0) = \sum_{j = 1}^{m} a_{j m} w_{j} (x) \to u_{0} (x) \in H_{0}^{2} (Ω), \\ (3.3) & u_{m t} (x, 0) = \sum_{j = 1}^{m} b_{j m} w_{j} (x) \to u_{1} (x) \in L^{2} (Ω), \end{array}$ where $a (u, v) = (Δ u, Δ v)$ and $b (u, v) = \sum_{i = 1}^{n} (σ_{i} (u_{x_{i}}), v_{x_{i}})$ .

Next, we give the following lemma to help prove the asymptotic behavior of global solutions to problem (1.1) for $E (0) < d_{1}$ .
Lemma 3.1.
Let $σ (s)$ satisfy (H) and ( 1.2 ) hold. Assume that $0 < E (0) < d_{1}$ and $u_{0} (x) \in W^{'}$ , where $d_{1}$ is defined by ( 2.5 ). Then for the approximate solutions $u_{m}$ to problem ( 1.1 ) given above, there hold
For any m, $\begin{matrix} (3.4) & I (u_{m}) = ‖ u_{m t} ‖^{2} - \frac{d}{d t} ((u_{m t}, u_{m}) + \frac{γ}{2} ‖ u_{m} ‖^{2}), 0 ⩽ t < + \infty; \end{matrix}$

There exists a constant $δ > 0$ such that $\begin{matrix} (3.5) & I (u_{m}) ⩾ δ ‖ Δ u_{m} ‖^{2} \end{matrix}$ for sufficiently large m and $0 ⩽ t < + \infty$ .

Proof.
Multiplying (3.1) by $g_{s m}^{'} (t)$ and summing for s gives $\begin{matrix} (3.6) & \frac{d}{d t} E_{m} (t) + γ ‖ u_{m t} ‖^{2} = 0 \end{matrix}$ and $\begin{matrix} (3.7) & E_{m} (t) + γ \int_{0}^{t} ‖ u_{m τ} ‖^{2} d τ = E_{m} (0), \end{matrix}$ where $E_{m} (t) = \frac{1}{2} ‖ u_{m t} ‖^{2} + J (u_{m})$ .

Then the two conclusions of this lemma can be proved as follows.
Multiplying (3.1) by $g_{s m} (t)$ and summing for s gives (3.4).

From $E (0) < d_{1}$ it follows that there exits a $δ \in (0, 1)$ such that $E (0) < {(1 - δ)}^{\frac{2}{q - 1}} d_{1}$ , which together with (3.2) and (3.3) shows $E_{m} (0) < {(1 - δ)}^{\frac{2}{q - 1}} d_{1}$ for sufficiently large m. Thus from (3.7) and (2.5) it implies $\begin{array}{l} \frac{1}{2} ‖ u_{m t} ‖^{2} + \frac{p - 1}{2 (p + 1)} ‖ Δ u_{m} ‖^{2} + \frac{1}{p + 1} I (u_{m}) < {(1 - δ)}^{\frac{2}{q - 1}} d_{1} = \frac{p - 1}{2 (p + 1)} {(\frac{1 - δ}{a C_{}})}^{\frac{2}{q - 1}}, \end{array}$ which gives $\begin{matrix} (3.8) & ‖ Δ u_{m} ‖ < {(\frac{1 - δ}{a C_{}})}^{\frac{1}{q - 1}} \end{matrix}$ for sufficiently large m. From assumption (H) and (3.8) we get $\begin{array}{l} \sum_{i = 1}^{n} \int_{Ω} u_{m x_{i}} σ_{i} (u_{m x_{i}}) d x ⩽ a ‖ \nabla u_{m} ‖_{q + 1}^{q + 1} ⩽ a C_{*} ‖ Δ u_{m} ‖^{q + 1} < (1 - δ) ‖ Δ u_{m} ‖^{2} \end{array}$ and $\begin{array}{l} I (u_{m}) & = ‖ Δ u_{m} ‖^{2} - \sum_{i = 1}^{n} \int_{Ω} u_{m x_{i}} σ_{i} (u_{m x_{i}}) d x \\ = δ ‖ Δ u_{m} ‖^{2} + (1 - δ) ‖ Δ u_{m} ‖^{2} - \sum_{i = 1}^{n} \int_{Ω} u_{m x_{i}} σ_{i} (u_{m x_{i}}) d x ⩾ δ ‖ Δ u_{m} ‖^{2} \end{array}$ for $0 ⩽ t < \infty$ and sufficiently large m.
Thus the proof is completed. □

Now we present the main conclusion in this subsection, that is, the asymptotic behavior for problem (1.1) with $E (0) < d_{1}$ as follows. Theorem 3.2 (Asymptotic behavior for $E (0) < d_{1}$ ).

Let $γ > 0$ , $σ (s)$ satisfy (H) and ( 1.2 ) hold. Assume that $0 < E (0) < d_{1}$ and $u_{0} (x) \in W^{'}$ . Then for global weak solutions to problem ( 1.1 ) given in Theorem 3.1 there exist positive constants C and λ such that $\begin{matrix} (3.9) & ‖ u_{t} ‖^{2} + ‖ Δ u ‖^{2} ⩽ C e^{- λ t}, 0 ⩽ t < \infty . \end{matrix}$

Proof.
Let ${u_{m}}$ be the approximate solutions to problem (1.1) defined before, then (3.6) holds. Multiplying (3.6) by $e^{α t} (α > 0)$ yields $\begin{matrix} \frac{d}{d t} (e^{α t} E_{m} (t)) + e^{α t} γ ‖ u_{m t} ‖^{2} = α e^{α t} E_{m} (t), 0 ⩽ t < \infty \end{matrix}$ and $\begin{matrix} (3.10) & e^{α t} E_{m} (t) + γ \int_{0}^{t} e^{α τ} ‖ u_{m τ} ‖^{2} d τ = E_{m} (0) + α \int_{0}^{t} e^{α τ} E_{m} (τ) d τ, 0 ⩽ t < \infty . \end{matrix}$ From (ii) in assumption (H), Lemma 2.1 and (3.8) it follows that $\begin{array}{l} - \sum_{i = 1}^{n} \int_{Ω} G_{i} (u_{m x_{i}}) d x ⩽ & \sum_{i = 1}^{n} \int_{Ω} | G_{i} (u_{m x_{i}}) | d x \\ ⩽ & \sum_{i = 1}^{n} \int_{Ω} \frac{a}{p + 1} | u_{m x_{i}} |^{q + 1} d x \\ = & \frac{a}{p + 1} ‖ \nabla u_{m} ‖_{q + 1}^{q + 1} \\ ⩽ & \frac{a C_{}}{p + 1} ‖ Δ u_{m} ‖^{q + 1} \\ = & \frac{a C_{}}{p + 1} ‖ Δ u_{m} ‖^{q - 1} ‖ Δ u_{m} ‖^{2} ⩽ \frac{1 - δ}{p + 1} ‖ Δ u_{m} ‖^{2}, 0 ⩽ t < + \infty \end{array}$ for all m. Hence by (3.4) and (3.5) we have $\begin{array}{l} (3.11) & \begin{matrix} \int_{0}^{t} e^{α τ} E_{m} (τ) d τ & = \int_{0}^{t} e^{α τ} (\frac{1}{2} ‖ u_{m τ} ‖^{2} + \frac{1}{2} ‖ Δ u_{m} ‖^{2} - \sum_{i = 1}^{n} \int_{Ω} G_{i} (u_{m x_{i}}) d x) d τ \\ ⩽ \int_{0}^{t} e^{α τ} (\frac{1}{2} ‖ u_{m τ} ‖^{2} + (\frac{1}{2} + \frac{1 - δ}{p + 1}) ‖ Δ u_{m} ‖^{2}) d τ \\ ⩽ \int_{0}^{t} e^{α τ} (\frac{1}{2} ‖ u_{m τ} ‖^{2} + C (p, δ) I (u_{m})) d τ \\ = (\frac{1}{2} + C (p, δ)) \int_{0}^{t} e^{α τ} ‖ u_{m τ} ‖^{2} d τ \\ - C (p, δ) \int_{0}^{t} e^{α τ} \frac{d}{d τ} ((u_{m τ}, u_{m}) + \frac{γ}{2} ‖ u_{m} ‖^{2}) d τ, 0 ⩽ t < + \infty \end{matrix} \end{array}$ for sufficiently large m, where $\begin{matrix} C (p, δ) = \frac{1}{δ} (\frac{1}{2} + \frac{1 - δ}{p + 1}) . \end{matrix}$ Further $\begin{array}{l} (3.12) & \begin{array}{l} - \int_{0}^{t} e^{α τ} \frac{d}{d τ} ((u_{m τ}, u_{m}) + \frac{γ}{2} ‖ u_{m} ‖^{2}) d τ \\ = (u_{m t} (0), u_{m} (0)) + \frac{γ}{2} {‖ u_{m} (0) ‖}^{2} - e^{α t} ((u_{m t}, u_{m}) + \frac{γ}{2} ‖ u_{m} ‖^{2}) \\ + α \int_{0}^{t} e^{α τ} ((u_{m τ}, u_{m}) + \frac{γ}{2} ‖ u_{m} ‖^{2}) d τ \\ ⩽ \frac{1}{2} ({‖ u_{m t} (0) ‖}^{2} + (1 + γ) {‖ u_{m} (0) ‖}^{2}) + \frac{1}{2} e^{α t} (‖ u_{m t} ‖^{2} + (1 + γ) ‖ u_{m} ‖^{2}) \\ + \frac{α}{2} \int_{0}^{t} e^{α τ} (‖ u_{m τ} ‖^{2} + (1 + γ) ‖ u_{m} ‖^{2}) d τ, 0 ⩽ t < + \infty . \end{array} \end{array}$ Moreover from the proof of Theorem 3.3 in [16] it follows that $u_{m} (t) \in W$ for $0 ⩽ t < + \infty$ and sufficiently large m. Hence by $\begin{array}{l} \frac{1}{2} ‖ u_{m t} ‖^{2} + \frac{p - 1}{2 (p + 1)} ‖ Δ u_{m} ‖^{2} + \frac{1}{p + 1} I (u_{m}) ⩽ \frac{1}{2} ‖ u_{m t} ‖^{2} + J (u_{m}) = E_{m} (t), 0 ⩽ t < + \infty, \end{array}$ we obtain $\begin{matrix} (3.13) & \frac{1}{2} ‖ u_{m t} ‖^{2} + \frac{p - 1}{2 (p + 1)} ‖ Δ u_{m} ‖^{2} ⩽ E_{m} (t), 0 ⩽ t < + \infty . \end{matrix}$ Thus there exists a positive constant C such that $\begin{matrix} (3.14) & \frac{1}{2} (‖ u_{m t} ‖^{2} + (1 + γ) ‖ u_{m} ‖^{2}) ⩽ C E_{m} (t), 0 ⩽ t < + \infty \end{matrix}$ for sufficiently large m. From (3.10)–(3.14) it follows that there exist positive constants $C_{0}$ and $C_{1}$ such that $\begin{array}{l} e^{α t} E_{m} (t) + γ \int_{0}^{t} e^{α τ} ‖ u_{m τ} ‖^{2} d τ \\ ⩽ C_{0} E_{m} (0) + α (\frac{1}{2} + C (p, δ)) \int_{0}^{t} e^{α τ} ‖ u_{m τ} ‖^{2} d τ \\ (3.15) & + α C_{1} e^{α t} E_{m} (t) + α^{2} C_{1} \int_{0}^{t} e^{α τ} E_{m} (τ) d τ, 0 ⩽ t < + \infty \end{array}$ for sufficiently large m. Taking α such that $\begin{matrix} 0 < α < min {\frac{1}{2 C_{1}}, \frac{γ}{\frac{1}{2} + C (p, δ)}}, \end{matrix}$ then (3.15) gives $\begin{array}{l} e^{α t} E_{m} (t) ⩽ 2 C_{0} E_{m} (0) + 2 α^{2} C_{1} \int_{0}^{t} e^{α τ} E_{m} (τ) d τ < 2 C_{0} d_{1} + 2 α^{2} C_{1} \int_{0}^{t} e^{α τ} E_{m} (τ) d τ . \end{array}$ In virtue of the Gronwall inequality we get $\begin{matrix} e^{α t} E_{m} (t) ⩽ 2 C_{0} d_{1} e^{2 C_{1} α^{2} t}, \end{matrix}$ that is $\begin{matrix} (3.16) & E_{m} (t) ⩽ 2 C_{0} d_{1} e^{- λ t}, 0 ⩽ t < + \infty \end{matrix}$ for sufficiently large m, where $λ = α (1 - 2 C_{1} α) > 0$ .

Let ${u_{ν}}$ be the subsequence of ${u_{m}}$ (see the proof of Theorem 4.1 in [16]), then by (3.13) and (3.16) we have $\begin{array}{l} ‖ u_{t} ‖^{2} + ‖ Δ u ‖^{2} & ⩽ \underset{ν \to \infty}{lim inf} (‖ u_{ν t} ‖^{2} + ‖ Δ u_{ν} ‖^{2}) \\ ⩽ \underset{ν \to \infty}{lim inf} \frac{2 (p + 1)}{p - 1} E_{ν} (t) \\ = \frac{4 (p + 1)}{p - 1} C_{0} d_{1} e^{- λ t}, 0 ⩽ t < + \infty . \end{array}$ Thus the proof of this theorem is completed. □

3.2. Finite time blow up for $E (0) < d$

Theorem 3.3 (Finite time blow up for $E (0) < d$ ).

Let $σ (s)$ satisfy (H) and ( 1.2 ) hold. Assume that $E (0) < d$ and $u_{0} (x) \in V^{'}$ . Then problem ( 1.1 ) does not admit any global weak solution.

Proof.
Let $u (t)$ be a weak solution to problem (1.1) with $E (0) < d$ , $u_{0} (x) \in V^{'}$ and T be the maximum existence time of $u (t)$ . Let us prove $T < + \infty$ . Arguing by contradiction, we suppose that $T = + \infty$ . Set $\begin{array}{l} (3.17) & F (t) : = ‖ u ‖^{2} + γ \int_{0}^{t} ‖ u ‖^{2} d τ + (T_{0} - t) γ ‖ u_{0} ‖^{2}, 0 < t < + \infty, \end{array}$ directly $\begin{array}{l} (3.18) & F^{'} (t) = 2 (u_{t}, u) + γ ‖ u ‖^{2} - γ ‖ u_{0} ‖^{2} = 2 (u_{t}, u) + 2 γ \int_{0}^{t} (u_{τ}, u) d τ, 0 < t < + \infty \end{array}$ and $\begin{array}{l} (3.19) & F^{″} (t) = 2 ‖ u_{t} ‖^{2} + 2 (u_{t t}, u) + 2 γ (u_{t}, u) = 2 ‖ u_{t} ‖^{2} - 2 I (u), 0 < t < + \infty . \end{array}$ Further from Schwartz inequality it implies $\begin{array}{l} {(u_{t}, u)}^{2} ⩽ ‖ u_{t} ‖^{2} ‖ u ‖^{2}, \\ {(γ \int_{0}^{t} (u_{τ}, u) d τ)}^{2} ⩽ γ^{2} \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ \int_{0}^{t} ‖ u ‖^{2} d τ \end{array}$ and $\begin{array}{l} 2 γ (u_{t}, u) \int_{0}^{t} (u_{τ}, u) d τ \\ ⩽ 2 γ ‖ u_{t} ‖ ‖ u ‖ {(\int_{0}^{t} ‖ u_{τ} ‖^{2} d τ)}^{\frac{1}{2}} {(\int_{0}^{t} ‖ u ‖^{2} d τ)}^{\frac{1}{2}} \\ ⩽ γ ‖ u ‖^{2} \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ + γ ‖ u_{t} ‖^{2} \int_{0}^{t} ‖ u ‖^{2} d τ, \end{array}$ which tells $\begin{array}{l} (3.20) & \begin{matrix} {(F^{'} (t))}^{2} & ⩽ 4 {((u_{t}, u) + γ \int_{0}^{t} (u_{τ}, u) d τ)}^{2} \\ = 4 ({(u_{t}, u)}^{2} + {(γ \int_{0}^{t} (u_{τ}, u) d τ)}^{2} + 2 γ (u_{t}, u) \int_{0}^{t} (u_{τ}, u) d τ) \\ ⩽ 4 (‖ u ‖^{2} + γ \int_{0}^{t} ‖ u ‖^{2} d τ) (‖ u_{t} ‖^{2} + γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ) \\ = 4 F (t) (‖ u_{t} ‖^{2} + γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ), 0 < t ⩽ T_{0} . \end{matrix} \end{array}$ So we have $\begin{array}{l} F (t) F^{″} (t) - \frac{p + 3}{4} {(F^{'} (t))}^{2} \\ ⩾ F (t) (2 ‖ u_{t} ‖^{2} - 2 I (u) - (p + 3) ‖ u_{t} ‖^{2} - (p + 3) γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ), \\ (3.21) & = F (t) (- (p + 1) ‖ u_{t} ‖^{2} - 2 I (u) - (p + 3) γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ), 0 < t ⩽ T_{0} . \end{array}$ Further from (2.7) and (2.1) it shows $\begin{array}{l} \frac{1}{2} ‖ u_{t} ‖^{2} + J (u) + γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ ⩽ E (0) \end{array}$ and $\begin{matrix} (3.22) & - (p + 1) ‖ u_{t} ‖^{2} ⩾ 2 (p + 1) (J (u) - E (0)) + 2 (p + 1) γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ, 0 < t ⩽ T_{0} . \end{matrix}$ Substituting (3.22) into (3.21) yields $\begin{array}{l} F (t) F^{″} (t) - \frac{p + 3}{4} {(F^{'} (t))}^{2} \\ ⩾ F (t) (2 (p + 1) (J (u) - E (0)) - 2 I (u) + (p - 1) γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ) \\ ⩾ 2 F (t) ((p + 1) (J (u) - E (0)) - I (u)), 0 < t ⩽ T_{0} . \end{array}$ From Theorem 2.1 it follows $u (t) \in V^{'}$ for $0 ⩽ t < + \infty$ . And Lemma 2.2 gives $‖ u ‖ > 0$ for $0 ⩽ t < + \infty$ . Further, from Lemma 2.5 and $E (0) < d$ we have $\begin{matrix} (p + 1) (J (u) - E (0)) - I (u) > 0 \end{matrix}$ and $\begin{matrix} (3.23) & F (t) F^{″} (t) - \frac{p + 3}{4} {(F^{'} (t))}^{2} > 0, 0 < t < T_{0} . \end{matrix}$

On the other hand, both (3.19) and (2.6) give $\begin{array}{l} F^{″} (t) ⩾ - 2 I (u) > 2 (p + 1) (d - J (u)) ⩾ 2 (p + 1) (d - E (0)) = δ_{0} > 0, 0 < t < + \infty \end{array}$ and $\begin{matrix} F^{'} (t) ⩾ δ_{0} t + F^{'} (0) = δ_{0} t + 2 (u_{0}, u_{1}), 0 < t < + \infty . \end{matrix}$ Hence there exists a $t_{0} ⩾ 0$ dependent only on $u_{0}$ and $u_{1}$ , but independent of $T_{0}$ such that $\begin{matrix} F^{'} (t_{0}) > \frac{4}{p - 1} γ ‖ u_{0} ‖^{2} \end{matrix}$ and $F (t_{0}) > 0$ for $T_{0} > t_{0}$ . Now we define $T_{0}$ by $\begin{matrix} \frac{‖ u (t_{0}) ‖^{2} + γ \int_{0}^{t_{0}} ‖ u ‖^{2} d t + (T_{0} - t_{0}) γ ‖ u_{0} ‖^{2}}{\frac{p - 1}{4} F^{'} (t_{0})} = T_{0} - t_{0} . \end{matrix}$ Then $\begin{matrix} T_{0} - t_{0} = \frac{‖ u (t_{0}) ‖^{2} + γ \int_{0}^{t_{0}} ‖ u ‖^{2} d t}{\frac{p - 1}{4} F^{'} (t_{0}) - γ ‖ u_{0} ‖^{2}} > 0 . \end{matrix}$ According to the convexity method [11,12], we can conclude that there exists a $T_{1} > t_{0}$ ensuring $\begin{matrix} T_{1} - t_{0} < \frac{F (t_{0})}{\frac{p - 1}{4} F^{'} (t_{0})} = \frac{‖ u (t_{0}) ‖^{2} + γ \int_{0}^{t_{0}} ‖ u ‖^{2} d t + (T_{0} - t_{0}) γ ‖ u_{0} ‖^{2}}{\frac{p - 1}{4} F^{'} (t_{0})} = T_{0} - t_{0} \end{matrix}$ such that $\begin{matrix} lim_{t \to T_{1}} F (t) = + \infty, \end{matrix}$ which contradicts $T_{1} < T_{0} < T = + \infty$ . This theorem now is proved. □

From Theorem 3.3 and (iii) of Theorem 2.1 we can obtain the following corollary.
Corollary 3.1.
Let $σ (s)$ satisfy (H) and ( 1.2 ) hold. Assume that $E (0) < 0$ or $E (0) = 0$ with $u_{0} \neq 0$ . Then problem ( 1.1 ) does not admit any global weak solution.

From Theorem 3.1 and Theorem 3.3 we can obtain a sharp condition for global existence and nonexistence of solutions to problem (1.1) for $0 < E (0) < d$ . Theorem 3.4.
Let $σ (s)$ satisfy (H) and ( 1.2 ) hold. Assume that $0 < E (0) < d$ . Then when $I (u_{0}) > 0$ problem ( 1.1 ) admits a global weak solution; and when $I (u_{0}) < 0$ problem ( 1.1 ) does not admit any global weak solution.

4. Finite time blow up for arbitrarily positive initial energy

This section deals with the finite time blow up of weak solutions to problem (1.1) for arbitrarily positive initial energy $E (0) > 0$ . We first show the following lemma to rebuild the corresponding invariant manifold.

Lemma 4.1.
Let $γ ⩾ 0$ , $σ (s)$ satisfy (H) and ( 1.2 ) hold. Then all weak solutions to problem ( 1.1 ) belong to $V^{'}$ , provided that $\begin{matrix} (4.1) & u_{0} \in V^{'} \end{matrix}$ and $\begin{matrix} (4.2) & (u_{0}, u_{1}) > \frac{2 δ_{1} (p + 1)}{η (p - 1)} E (0) > 0, \end{matrix}$ where $δ_{1} = max {γ, 1}$ , $η = min {λ_{1}^{2}, 1}$ and $\begin{matrix} λ_{1} = inf_{u \in H_{0}^{2} (Ω)} \frac{‖ Δ u ‖}{‖ u ‖} . \end{matrix}$
Proof.
We claim that $u \in V^{'}$ . Arguing by contradiction, due to the continuity of $I (t)$ in t, we suppose that $t_{0} \in (0, T)$ is the first time such that $\begin{array}{l} (4.3) & I (u (t_{0})) = 0 \end{array}$ and $\begin{array}{l} (4.4) & I (u (t)) < 0 for t \in [0, t_{0}) . \end{array}$ First we define $\begin{array}{l} θ (t) : = γ {‖ u (t) ‖}^{2} + 2 (u, u_{t}) . \end{array}$ Testing the equation in problem (1.1) by $u (t)$ and from (2.3) we have $\begin{array}{l} (4.5) & θ^{'} (t) = 2 γ (u, u_{t}) + 2 (u_{t t}, u) + 2 ‖ u_{t} ‖^{2} = 2 ‖ u_{t} ‖^{2} - 2 I (u) . \end{array}$ Hence by (4.5) and (4.4) we get $\begin{array}{l} (4.6) & θ^{'} (t) > 0 for t \in [0, t_{0}) . \end{array}$ In fact, from (4.2) it implies that $\begin{array}{l} (4.7) & θ (0) = γ ‖ u_{0} ‖^{2} + 2 (u_{0}, u_{1}) > 2 (u_{0}, u_{1}) > \frac{4 (p + 1)}{η (p - 1)} E (0) > 0 . \end{array}$ Therefore from (4.6) and (4.7) it yields $\begin{array}{l} (4.8) & θ (t) > θ (0) > 0 for t \in [0, t_{0}) . \end{array}$ Next we claim that $\begin{array}{l} (4.9) & H (t) : = \int_{Ω} u u_{t} d x > 0 for t \in [0, t_{0}) . \end{array}$ Indeed it is easy to see that $\begin{array}{l} H^{'} (t) = (u, u_{t t}) + ‖ u_{t} ‖^{2} = - I (u) - γ (u_{t}, u) + ‖ u_{t} ‖^{2}, \end{array}$ which together with (4.4) shows $\begin{matrix} (4.10) & H^{'} (t) + γ H (t) = - I (u) + ‖ u_{t} ‖^{2} > 0 for t \in [0, t_{0}) . \end{matrix}$ A simple computation on (4.10) shows $H (t) e^{γ t} > H (0) for t \in [0, t_{0})$ , which together with (4.2) gives (4.9).

Now by (4.9), (4.8) and (4.2) we can conclude that $\begin{array}{l} (4.11) & \begin{array}{l} δ_{1} ({‖ u (t) ‖}^{2} + 2 (u (t), u_{t} (t))) \\ ⩾ γ {‖ u (t) ‖}^{2} + 2 (u (t), u_{t} (t)) \\ > γ ‖ u_{0} ‖^{2} + 2 (u_{0}, u_{1}) > 2 (u_{0}, u_{1}) > \frac{4 δ_{1} (p + 1)}{η (p - 1)} E (0) for all t \in [0, t_{0}), \end{array} \end{array}$ that is $\begin{array}{l} {‖ u (t) ‖}^{2} + 2 (u (t), u_{t} (t)) > \frac{4 (p + 1)}{η (p - 1)} E (0) for all t \in (0, t_{0}) . \end{array}$ Due to the continuity of $u (t)$ and $u_{t} (t)$ in t it gives $\begin{matrix} (4.12) & {‖ u (t_{0}) ‖}^{2} + 2 (u (t_{0}), u_{t} (t_{0})) > \frac{4 (p + 1)}{η (p - 1)} E (0) . \end{matrix}$ Recalling (2.1), (2.2), (2.3) and (2.4) we have $\begin{array}{l} (4.13) & \begin{matrix} E (t) = & \frac{1}{2} ‖ u_{t} ‖^{2} + \frac{1}{2} ‖ Δ u ‖^{2} - \sum_{i = 1}^{n} \int_{Ω} G_{i} (u_{x_{i}}) d x \\ = & \frac{1}{2} ‖ u_{t} ‖^{2} + (\frac{1}{2} - \frac{1}{p + 1}) ‖ Δ u ‖^{2} + \frac{1}{p + 1} \sum_{i = 1}^{n} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x \\ - \sum_{i = 1}^{n} \int_{Ω} G_{i} (u_{x_{i}}) d x + \frac{1}{p + 1} (‖ Δ u ‖^{2} - \sum_{i = 1}^{n} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x) \\ = & \frac{1}{2} ‖ u_{t} ‖^{2} + (\frac{1}{2} - \frac{1}{p + 1}) ‖ Δ u ‖^{2} + \frac{1}{p + 1} I (u) \\ + \frac{1}{p + 1} \sum_{i = 1}^{n} \int_{Ω} u_{x_{i}} σ_{i} (u_{x_{i}}) d x - \sum_{i = 1}^{n} \int_{Ω} G_{i} (u_{x_{i}}) d x \\ ⩾ & \frac{1}{2} ‖ u_{t} ‖^{2} + (\frac{1}{2} - \frac{1}{p + 1}) ‖ Δ u ‖^{2} + \frac{1}{p + 1} I (u), \end{matrix} \end{array}$ which together with (2.7), (4.3) and Cauchy–Schwarz inequality yields $\begin{array}{l} E (0) & > \frac{1}{2} {‖ u_{t} (t_{0}) ‖}^{2} + (\frac{1}{2} - \frac{1}{p + 1}) {‖ Δ u (t_{0}) ‖}^{2} \\ > \frac{p - 1}{4 (p + 1)} (2 {‖ u_{t} (t_{0}) ‖}^{2} + 2 {‖ Δ u (t_{0}) ‖}^{2}) \\ > \frac{p - 1}{4 (p + 1)} (2 {‖ u_{t} (t_{0}) ‖}^{2} + 2 λ_{1}^{2} {‖ u (t_{0}) ‖}^{2}) \\ > \frac{η (p - 1)}{4 (p + 1)} ({‖ u_{t} (t_{0}) ‖}^{2} + 2 {‖ u (t_{0}) ‖}^{2}) \\ > \frac{η (p - 1)}{4 (p + 1)} (2 (u_{t} (t_{0}), u (t_{0})) + {‖ u (t_{0}) ‖}^{2}) . \end{array}$ Hence we can derive $\begin{matrix} {‖ u (t_{0}) ‖}^{2} + 2 (u (t_{0}), u_{t} (t_{0})) < \frac{4 (p + 1)}{η (p - 1)} E (0), \end{matrix}$ which leads to a contradiction to (4.12). Therefore, this theorem is complete. □
Theorem 4.1 (Finite time blow up for $E (0) > 0$ ).

Let $γ ⩾ 0$ , $σ (s)$ satisfy (H) and ( 1.2 ) hold. Assume that ( 4.1 ) and ( 4.2 ) hold, then all solutions to problem ( 1.1 ) blow up in finite time.

Proof.

Let $u (t)$ be a weak solution to problem (1.1) with (4.1), (4.2) and T be the maximum existence time of $u (t)$ . From Lemma 4.1 it follows $u (t) \in V^{'}$ . Now we claim $T < \infty$ . Arguing by contradiction, we suppose that $T = + \infty$ . Recalling the auxiliary function $F (t)$ defined by (3.17), we have (3.18), (3.19) and (3.20), which gives $\begin{array}{l} F (t) F^{″} (t) - \frac{β + 3}{4} {(F^{'} (t))}^{2} \\ ⩾ F (t) (2 ‖ u_{t} ‖^{2} - 2 I (u) - (β + 3) ‖ u_{t} ‖^{2} - (β + 3) γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ) \\ = F (t) (- (β + 1) ‖ u_{t} ‖^{2} - 2 I (u) - (β + 3) γ \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ), \end{array}$ where $p > β > 1$ will be decided later. Let $\begin{matrix} ξ (t) : = - (β + 1) ‖ u_{t} ‖^{2} - 2 I (u) - γ (β + 3) \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ, \end{matrix}$ which together with (4.13) and (2.7) becomes $\begin{array}{l} (4.14) & \begin{matrix} ξ (t) = & (p - β) ‖ u_{t} ‖^{2} + (p - 1) ‖ Δ u ‖^{2} - 2 (p + 1) E (t) - γ (β + 3) \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ \\ = & (p - β) ‖ u_{t} ‖^{2} + (p - 1) ‖ Δ u ‖^{2} - 2 (p + 1) E (0) + γ (2 p - 1 - β) \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ \\ ⩾ & (p - β) ‖ u_{t} ‖^{2} + (p - 1) λ_{1}^{2} ‖ u ‖^{2} - 2 (p + 1) E (0) + γ (2 p - 1 - β) \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ . \end{matrix} \end{array}$ Then by taking $β = \frac{p + 1}{2}$ and Cauchy–Schwarz inequality, (4.14) becomes $\begin{array}{l} (4.15) & \begin{matrix} ξ (t) ⩾ & \frac{p - 1}{2} ‖ u_{t} ‖^{2} + (p - 1) λ_{1}^{2} ‖ u ‖^{2} - 2 (p + 1) E (0) + \frac{3 γ (p - 1)}{2} \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ \\ ⩾ & \frac{η (p - 1)}{2} (‖ u_{t} ‖^{2} + 2 ‖ u ‖^{2}) - 2 (p + 1) E (0) + \frac{3 γ (p - 1)}{2} \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ \\ ⩾ & \frac{η (p - 1)}{2} (2 (u, u_{t}) + ‖ u ‖^{2}) - 2 (p + 1) E (0) + \frac{3 γ (p - 1)}{2} \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ . \end{matrix} \end{array}$ Again recalling the proof of Lemma 4.1 we also find that both $2 (u, u_{t}) + ‖ u ‖^{2}$ and $(u, u_{t})$ are positive and $2 (u, u_{t}) + ‖ u ‖^{2}$ increases strictly over $[0, T)$ provided that $u (t) \in V^{'}$ . At this point, by (4.2) we can deduce from (4.15) that $\begin{array}{l} (4.16) & \begin{matrix} ξ (t) > & \frac{η (p - 1)}{2} (2 (u_{0}, u_{1}) + ‖ u_{0} ‖^{2}) - 2 (p + 1) E (0) + \frac{3 γ (p - 1)}{2} \int_{0}^{t} ‖ u_{τ} ‖^{2} d τ \\ > & η (p - 1) (u_{0}, u_{1}) - 2 (p + 1) E (0) : = \bar{δ_{0}} > 0, t \in (0, T_{0}), \end{matrix} \end{array}$ which implies $\begin{matrix} F (t) F^{″} (t) - \frac{β + 3}{4} {(F^{'} (t))}^{2} > C (\bar{δ_{0}}) > 0 . \end{matrix}$ The reminder of this proof is similar to that of Theorem 3.3 and follows from the convexity method [11,12]. □

Footnotes

Acknowledgements

The work is partially supported by the National Natural Science Foundation of China (11871017, 11801114) and the Fundamental Research Funds for the Central Universities.

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Asymptotic behavior and finite time blow up for damped fourth order nonlinear evolution equation

Abstract

Keywords

1. Introduction

1.1. Background

1.2. Motivation

1.3. Organization

2. Preliminaries

Lemma 2.3 ([16]).

3.1. Global existence and asymptotic behavior for E ( 0 ) < d

Theorem 3.3 (Finite time blow up for E ( 0 ) < d ).

Footnotes

Acknowledgements

References

3.1. Global existence and asymptotic behavior for $E (0) < d$

Theorem 3.3 (Finite time blow up for $E (0) < d$ ).