THE EXISTENCE, GENERAL DECAY AND BLOW-UP FOR A PLATE EQUATION WITH NONLINEAR DAMPING AND A LOGARITHMIC SOURCE TERM

In this paper, we consider a plate equation with nonlinear damping and logarithmic source term. By the contraction mapping principle, we establish the local existence. The global existence and decay estimate of the solution at subcritical initial energy are obtained. We also prove that the solution with negative initial energy blows up in finite time under suitable conditions. Moreover, we also give the blow-up in finite time of solution at the arbitrarily high initial energy for linear damping (i.e. m = 2).


Introduction
In this paper, we deal with the following plate equation with nonlinear damping and a logarithmic source term where Ω is a bounded domain in R n (n ≥ 1) with sufficiently smooth boundary ∂Ω, ν is the unit outer normal to ∂Ω and k is a positive real number, u 0 (x), and u 1 (x) are given initial data. The parameter m ≥ 2 and p satisfies (2) 2 < p < 2(n − 2) n − 4 if n ≥ 5; 2 < p < +∞ if n ≤ 4.
The logarithmic nonlinearity is of much interest in many branches of physics such as nuclear physics, optics and geophysics (see [5,6,15] and references therein). It has also been applied in quantum field theory, where this kind of nonlinearity appears naturally in cosmological inflation and in super symmetric field theories [4,13].
Let us review somework with logarithmic term which is closely related to the problem (1). Birula and Mycielski [6,7] studied the following problem u(a, t) = u(b, t) = 0, t ∈ (0, T ), which is a relativistic version of logarithmic quantum mechanics and can also be obtained by taking the limit p → 1 for the p-adic string equation [16,36]. Cazenave and Haraux [8] established the existence and uniqueness of a solution to the Cauchy problem for the following equation (4) u tt − ∆u = u log |u| k , in R 3 . Using some compactness method, Gorka [15] established the global existence of weak solutions for all (u 0 , u 1 ) ∈ H 1 0 × L 2 to the initial boundary value problem of equation (4) in the one-dimensional case. In [5], Bartkowski and Gorka obtained the existence of classical solutions and investigated weak solutions for the corresponding Cauchy problem of equation (4) in the one-dimensional case. Recently, using potential well combined with logarithmic Sobolev inequality, Lian et al. [25] derived the global existence and infinite time blow up of the solution to the initial boundary value problem of (4) in finite dimensional case under suitable assumptions on initial data. Similar results were obtained by Lian et al. [26] for nonlinear wave equation with weak and strong damping terms and logarithmic source term. Hiramatsu et al. [19] also introduced the following equation (5) u tt − ∆u + u + u t + |u| 2 u = u log |u| to study the dynamics of Q-ball in theoretical physics. A numerical research was given in that work, while, there was no theoretical analysis for this problem. For the initial boundary value problem of(5), Han [17] obtained the global existence of weak solution in R 3 , and Zhang et al. [40] obtained the decay estimate of energy for the problem (5) in finite dimensional case. Later, the authors in [20] considered the initial boundary problem of (5) in Ω ⊂ R 3 , they proved that the solution will grow exponentially as time goes to infinity if the solution lies in unstable set or the initial energy is negative; the decay rate of the energy was also obtained if the solution lies in a smaller set compared with the stable set. Peyravi [35] extended the results obtained in [20] to the following logarithmic wave equation Recently, Al-Gharabli and Messaoudi [1] considered the following plate equation with logarithmic source term where Ω is a bounded domain in R 2 , and obtained the global existence and decay rate of the solution using the multiplier method. As the special case , i.e. h(u t ) = u t in (6), the same authors [2] established the global existence and the decay estimate by constructing a Lyapunov function. Moreover, Al-Gharabli et al. [3] considered the following initial boundary value problem of viscoelastic plate equation with with logarithmic source term they established the existence of solutions and proved an explicit and general decay rate result. However, there is no information on the finite or infinite blow up results in these researches [1,2,3]. At the same time, there are many results concerning the existence and nonexistence on evolution equation with polynomial source term. For example, for plate equation with polynomial source term |u| p−2 u, Messaoudi [31] considered the following problem established an existence result and showed that the solution continues to exist globally if m ≥ p and blows up in finite time if m < p and the initial energy is negative. This result was later improved by Chen and Zhou [12], see also Wu and Tsai [37].
Here, we also mention that there are a lot of results on the global well-posedness of solutions to the initial boundary value problem of nonlinear wave equations can be found [30,39] and papers cited therein by using of potential well method.
To the best of our knowledge, there are few results on the evolution equation with the nonlinear logarithmic source term |u| p−2 u log |u| k (p > 2). Kafini and Messaoudi [22] studied the following delayed wave equation with nonlinear logarithmic source obtained the local existence by using the semigroup theory and proved a finite time blow-up result when the initial energy is negative. Of course, these results also hold for the equation (8) without delay term (i.e. µ 2 = 0). However, there are no results on general decay and blow-up for positive initial energy. Chen et al. [9,10] studied parabolic type equations with logarithmic nonlinearity u log |u| k , and obtained the global existence of solution and the solutions cannot blow up in finite time.
Recently, Chen and Xu [11] study the initial-boundary value problem for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity and obtain the similar results. Nhan and Truong studied parabolic p-Laplacian equation [23] and pseudo-parabolic p-Laplacian equation [24] with logarithmic nonlinearity |u| p−2 u log |u| k where they need the p-Laplacian term to control the logarithmic nonlinearity. We also refer to [18], where pseudo-parabolic p-Laplacian equation with logarithmic nonlinearity |u| q−2 u log |u| k was considered. Motivated by the above mentioned papers, our purpose in this research is to investigative the existence, energy decay and finite time blow-up of the solution to the initial boundary value problem (1). We note here that (i) the term u plays an important role in the studying the problem (6)( see [1,2]) and (7) (see [3]) when the Logarithmic Sobolev inequality is used, while we do not care the term u in this paper; (ii) The constant k in (6) and (7) should satisfy 0 < k < k 0 , where k 0 is defined by 2π k0cp = e −3− 2 k 0 ( see details in [1,2,3]), while we only need k > 0.
The rest of this article is organized as follows: Section 2 is concerned with some notation and some properties of the potential well. In Sect. 3, we present the existence and uniqueness of local solutions to (1) by using the contraction mapping principle. In Sect. 4, we prove the global existence and energy decay results. The proof of global existence result is based on the potential well theory and the continuous principle; while for energy decay result, the proof is based on the Nakao's inequality and some techniques on logarithmic nonlinearity. In Sect. 5, we prove the the finite time blow-up when the initial energy is negative. In Sect. 6, we establish the finite time blow-up result for problem (1) with m = 2 under the arbitrarily high initial energy level (E(0) > 0).

Preliminaries
We give some material needed in the proof of our results. We use the standard Lebesgue space L p (Ω) and Sobolev space H 2 0 (Ω) with their usual scalar products and norms. In particular, we denote . = . 2 . By Poincaré 's inequality [27], we have that ∆ · is equivalent to · H 2 0 and we will use ∆ · as the norm of · H 2 0 , the corresponding duality between H 2 0 and H −2 is denote by ·, · . We also use C and C i to denote various positive constant that may have different values in different places.
Suppose (2) holds, we define for any α ∈ [0, α * ), then H 2 0 (Ω) → L p+α (Ω) continuously. And we denote C p+α by C * . From the definitions (11) and (12), we have The following lemmas play an important role in the studying the properties of the potential well.
Proof. We know It is obvious that (i) holds due to p ≥ 2 and u p = 0. Taking derivative of g(λ) we obtain From (14) and p ≥ 2, we see that there exists a unique positive λ * such that g (λ)| λ=λ * = 0, then we obtain Substituting the above equation into g (λ), we have From these and (i), we can yield that g(λ) has a maximum value at λ = λ * and J(λu) increasing on 0 < λ ≤ λ * and decreasing on λ * ≤ λ < +∞. So (ii) holds.
From (12) and (14), we have Then, we could define the potential well depth of the functional J (also known as mountain pass level ) by We also define the well-known Nehari manifold [29,34], that the mountain pass level d defined in (15) can also be characterized as It is easy to see that d ≥ 0 from (13). Now, we will prove that d is strictly positive.
Proof. Since log y < y for any constant y > 0, using (9) and the definition of I(u) in (12), we obtain that (16) Obviously, the results can be obtained from the above inequality (16).
where B = C p as in (9) and |Ω| is the measure of Ω.
Proof. It is obvious that r * > 0 (if exists), hence, we only need to prove r(α) ≤ γ(α), r * exists and r * < +∞, where For any u ∈ H 2 0 (Ω), using the Hölder's inequality, we have Then, noticing C * = C p+α and B = C p , we deduce Now, we will prove r * exists and r * < +∞. For this purpose, we divide the proof into two cases.
(ii) For the case p > 2, by (ii) of Corollary 2.1, we get that ∇u 2 ≥ r * if u ∈ N . Then, making using of (13) with I(u) > 0, we obtian We define energy for the problem (1), which obeys the following energy equality of the weak solution u It is obvious that Taking v = u t in (10), after a simple calculation, we get Now, we define the subsets of H 2 0 (Ω) related to problem (1). Set where W and V are called the stable and unstable set, respectively [21] . In order to establish the global existence and blow-up results of solution, we have to prove the following invariance sets of W and V .
Proof. It follows from the definition of weak solution and (17) that (i) Arguing by contradiction, we assume that there exists a number t 0 ∈ (0, T ) such that u(t) ∈ W on [0, t 0 ) and u (t 0 ) / ∈ W. Then, by the continuity of J(u(t)) and I(u(t)) with respect to t, we have either (a)J (u (t 0 )) = d or (b) I (u (t 0 )) = 0 and u (t 0 ) = 0.
It follows from (20) that (a) is impossible. If (b) holds, then by the definition of d, we have J(u(t 0 )) ≥ d, which contradicts (20) again. Thus, we have u(t) ∈ W for all t ∈ [0, T ).
(ii) The proof is similar to the proof of (i). We omit it.

Local existence
In this section, we are concerned with the local existence and uniqueness for the solution of the problem (1). The idea comes from [14,28,38], where the source term is polynomial. First, we give a technical lemma given in [22] which plays an important role in the uniqueness of the solution.  .
For every given u ∈ H, we consider the following initial boundary value problem We shall prove that the problem (21) admits a unique solution v ∈ H ∩ C 2 (|0, T |, is the orthogonal complete system of eigenfunctions of ∆ 2 in H 2 0 (Ω) with w i = 1 for all i. Then, {w i } is orthogonal and complete in L 2 (Ω) and in H 2 0 (Ω); denote by {λ i } the related eigenvalues to their multiplicity. Let solves the following problem For i = 1, . . . , h, taking η = w i in (23) yields the following Cauchy problem for a nonlinear ordinary differential equation with unknown γ ih Then the above problem admits a unique local solution γ ih ∈ C 2 [0, T ] for all i, which in turn implies a unique v h defined by (22) satisfying (23).
for every h ≥ 1. We estimate the last term in the right-hand side of (24) thanks to Young's and Sobolev's inequalities In order to estimate (25), we focus on the logarithmic term. Here we denote Ω := Ω 1 ∪ Ω 2 , where Ω 1 = {x ∈ Ω||u(x)| < 1} and Ω 2 = {x ∈ Ω||u(x)| ≥ 1}. Then we have The proof of the case n ≤ 4 is similar. From the above discussion, (25) yields Substituting this inequality into (24), we obtain where C > 0 is independent of h. It follows from (27) that Hence, up to a subsequence, we could pass to the limit in (23) and obtain a weak solution v of (21) with regularity (28). (21), we obtain v ∈ C 0 ([0, T ], H −2 (Ω)). Then the weak local solution of problem (21) has been obtained.
To prove the uniqueness, arguing by contradiction: if w and v were two solutions of (21) which have the same initial data. Subtracting these two equations and testing the result by w t − v t , we could obtain It follows from the following element inequality that (29) can make to be Therefore, we have w = v, i.e. the problem (21) obeys a unique weak solution. Now, we are in the position to prove Theorem 3.1. For u 0 ∈ H 2 0 (Ω), u 1 ∈ L 2 (Ω), we denote R 2 := 2 u 1 2 + ∆u 0 2 , and B RT := {u ∈ H|u(0, x) = u 0 (x), u t (0, x) = u 1 (x), u H ≤ R} for every T > 0. From the above discussion, for any u ∈ B RT , we could introduce a map Φ : H → H defined by v = Φ(u), where v is the unique solution to (21).
Claim. Φ is a contract map satisfying Φ (B RT ) ⊆ B RT , for small T > 0.
In fact, assume that u ∈ B RT , the corresponding solution v = Φ(u) satisfies (21) for all t ∈ [0, T ]. Thus, as did the proof (26) and (27) for n ≥ 5, For the case n ≤ 4, the index 2n n−4 of R in the last inequality can be replaced by any fixed positive number. If T is sufficiently small, then v H ≤ R, for any η ∈ H 2 0 (Ω) and a.e. t ∈ [0, T ].
and integrating both sides of (30) over (0, t), we have We need estimating the logarithmic term in (31) by using Lemma 3.1. By the similar argument as [22], we give the sketch of the proof. Making use of mean value theorem, we have, for 0 < θ < 1, Then, it follows from Lemma 3.1 that Since w 1 , w 2 ∈ B RT , using Hölder's inequality and the Sobolev embedding, we can obtain . By the similar argument, we have Using (2), we can choose sufficiently small ε > 0 such that which yields that (Ω) , from the above discussions, we can deduce that We choose T sufficiently small such that C R p−2 + 1 + Rp (p−2)/2(p−1) T < 1. Thus from (32) we obtain Φ is a contract map in B RT . The contraction mapping principle then shows that there exists a unique u ∈ B RT satisfying u = Φ(u) which is a solution to problem (1). The proof is complete.

Global existence and energy decay
In this section, we consider the global existence and energy decay of the solution for problem (1). First, we introduce the following lemmas which play an important role in studying the decay estimate of global solution for the problem (1).
where ω 0 is a positive constant and r is a nonnegative constant. Then we have where ω 1 = log w0 ω0−1 , here ω 0 > 1. Now, we establish the global existence and energy decay results.
Theorem 4.2. Let u be the unique local solution to problem (1). Assume (2) and 2 ≤ m < p hold. If u 0 ∈ W , u 1 ∈ L 2 (Ω) and E(0) < d, then u(t) is the global solution to the problem (1). Moreover it has the following decay property where K and κ are positive constants, τ is given by (47). Proof.
Step 1. Global existence. It suffices to show that u t 2 + ∆u 2 is uniformly bounded with respect to t. It follows from Lemma 2.5 (i) that u(t) ∈ W on [0, T ]. Using (13), we have the following estimate which yields that The above inequality and the continuation principle imply the global existence, i.e. T = +∞.
Step 2. We claim that there exists constant θ ∈ (0, 1) such that In fact, it follows from I(u(t)) > 0 for all t ≥ 0 and Lemma 2.1 that there exists a λ 0 > 1 such that I(λ 0 u(t)) = 0. Making use of (33), we have which implies that It follows from (12) that Combining this equality with (35), we have which implies that Hence, the inequality (34) holds with θ = 1 − λ 2−p 0 .

Blow up for negative energy
In this section, we will establish that the solution of problem (1) blows up in finite time provided E(0) < 0. For this purpose, we give some useful lemmas. for any u ∈ H 2 0 (Ω) and 2 ≤ s ≤ p, provided that Ω |u| p log |u| k dx ≥ 0. Lemma 5.2. Assume that (2) holds. Then there exists a positive constant C such that for any u ∈ H 2 0 (Ω), provided that Ω |u| p log |u| k dx ≥ 0.
Lemma 5.3. Assume that (2) holds. Then there exists a positive constant C > 1 such that u s p ≤ C u p p + ∇u 2 2 , for any u ∈ H 2 0 (Ω) and 2 ≤ s ≤ p. The proof of lemma 5.1-5.3 is similar to the proof in [22], we omit the details. Proof. We denote H(t) = −E(t). It follows from (17) and (18) that We define where ε > 0 to be determined later and By taking a derivation of L(t), we get Adding and subtracting εp(1 − a)H(t) for some a ∈ (0, 1) in the RHS of the above equation, then using the definition of H(t), we obtain In view of Young's inequality, we have for any δ > 0, which yields, by substitution in (50), Since the integral is taken over the x variable, (50) holds even if δ is time dependent. Thus by choosing δ so that δ −m/(m−1) = M H −β (t), for large M to be determined later, substituting in (51), we obtain Making using of (48), Lemma 5.4 and Young's inequality, we find Hence, it follows from Lemma 5.1 that Thus, Lemma 5.1 implies Combining (52) and (53), we have Now, we choose a > 0 sufficiently small that Thus, for some constant γ > 0, (54) has the form (55) L (t) ≥ γ H(t) + u t 2 + ∆u 2 + u p p + Ω |u| p log |u| k dx .
On the other hand, using Lemma 5.3, by the same method as in [32], we can deduce (56) L  where λ > 0 is constant depending only on γ and C. By a simple integration of (57) over (0, t), we have .
which implies that L(t) blow up in finite time This completes the proof of Theorem 5.1.

Arbitrarily high initial energy for linear damping
In this section, we consider the problem (1) with the linear damping term, i.e. m = 2. We will establish the finite time blow-up result by the method of the so called concavity method. For simplicity, we denote u 2 ≤ B 0 ∆u 2 . Lemma 6.1. [14] Let δ ≥ 0, T > 0 and h be a Lipschitzian function over [0, T ). Assume that h(0) ≥ 0 and h (t) + δh(t) > 0 for a.e. t ∈ [0, T ). Then h(t) > 0 for all t ∈ (0, T ).

Hence, we have
which is a contradiction with (61). The proof is complete.