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Global Well-posedness of Solutions for the p -Laplacian Hyperbolic Type Equation with Weak and Strong Damping Terms and Logarithmic Nonlinearity

. In this paper, we consider the p -Laplacian hyperbolic type equation with weak and strong damping terms and logarithmic nonlinearity. By using the potential well method and a logarithmic Sobolev inequality, we prove global existence, inﬁnite time blow up and asymptotic behavior of solutions in two cases E (0) < d and E (0) = d . Furthermore, the inﬁnite time blow up of solutions for the problem with E (0) > 0 ( ω = 0) is studied.


Introduction
In this paper, we study the initial boundary value problem of the p-Laplacian hyperbolic type equation with weak and strong damping terms and logarithmic nonlinearity where Ω ⊂ R n is a bounded domain with sufficiently smooth boundary ∂Ω, p > 2, ω ≥ 0, and ∆ p u = div(|∇u| p−2 ∇u) is the classical p-Laplacian operator.
The wave equation of p-Laplacian type has an active area of research and the most of the results are mainly concerned with the global well-posedness and stability of solutions. Messaoudi [24] considered the following quasilinear hyperbolic equations with nonlinear damping and source terms (1.2) u tt − div(|∇u| p−2 ∇u) − ∆u t + |u t | q−1 u t = |u| p−1 u, 2 Nouri Boumaza, Billel Gheraibia and Gongwei Liu and studied decay of solutions by using the techniques combination of the perturbed energy and potential well methods. Wu and Xue [31] studied (1.2) and proved the uniform energy decay rates of the solutions by utilizing the multiplier method. Pei et al. [26] studied the following quasilinear wave equation of p-Laplacian type u tt − div(|∇u| p−2 ∇u) − ∆u t = f (u), and proved the existence of local and global solutions. Also they proved the blow-up result of solutions with negative initial energy. Recently, Pereira et al. [27] considered the following quasilinear wave equation of p-Laplacian type with p-Laplacian damping They established the global solutions by means of the Faedo-Galerkin approximations and the asymptotic behavior is obtained by Nakao's method. For other related results, the readers may see [16,17,28] and the references therein. The logarithmic nonlinearity is of much interest in physics, since it appears naturally in inflation cosmology and super symmetric filed theories, quantum mechanics and nuclear physics [1,11]. Cazenave and Haraux [4] considered (1.3) u tt − ∆u = u ln |u| k in R 3 , and established the existence and uniqueness of the solution for the Cauchy problem. Górka [12] used some compactness arguments and obtained the global existence of weak solutions for all (u 0 , u 1 ) ∈ H 1 0 × L 2 to the initial-boundary value problem (1.3) in the one-dimensional case. Hiramatsu et al. [14] introduced the following equation when studying the dynamics of Q-balls in theoretical physics. In [13], Han proved the global existence of weak solutions, for all (u 0 , u 1 ) ∈ H 1 0 × L 2 to the initial boundary value problem (1.4) in Ω ⊂ R 3 . By constructing an appropriate Lyapunov function, Zhang and Liu [33] obtained the exponential decay estimates of energy with E(0) < d for all Recently, Lian and Xu [21] considered the initial boundary value problem for the following nonlinear wave equation and studied the well-posedness of the solution. Chen and Xu [7] considered the following wave equation Global Well-posedness of Solutions 3 and proved the global existence and infinite time blow up of solutions in cases of E(0) < d and E(0) = d. Also they proved the infinite time blow-up of solutions at arbitrarily high initial energy level (E(0) > 0) under two hypothetical situations. For the related works of equations with logarithmic nonlinearity, we also refer other works [2,5,6,8,10,15,20,22,23,32] and the references therein.
In recent years, the p-Laplacian equation with logarithmic nonlinearity have become an active area of research, see for example [3,10,19,25,29,30] and the references therein. Nhan and Truong [25] considered the p-Laplacian pseudo-parabolic equations with logarithmic nonlinearity and established the global existence, blow up and decay of solutions for p > 2. Cao and Liu [3] studied (1.5) and proved the existence and boundedness of global weak solutions, and also obtained the blow up at infinite time for 1 < p < 2. Pişkin and Irkil [30] considered the following equation with logarithmic nonlinearity They established the local existence of the solutions by using Faedo-Galerkin method. Pişkin et al. [29] studied the following p-Laplacian hyperbolic type equation with logarithmic nonlinearity u tt − div(|∇u| p−2 ∇u) − |u| p−2 u + u t = u p−2 u ln |u|.
They obtained the existence of global of solutions by the potential well theory and a logarithmic Sobolev inequality. Furthermore, the growth and the decay estimates of solutions for the equation are studied.
Motivated by the above mentioned papers, our purpose in this research is to investigative the global existence, asymptotic behavior and infinite time blow up of solution in cases E(0) < d and E(0) = d to the initial boundary value problem (1.1). Furthermore, the infinite time blow up of solutions for the problem (1.1) with E(0) > 0 is studied.
Our paper is organized as follows. In Section 2, we give some notations, preliminary lemmas, and introduce some potential wells. In Section 3, we prove the global existence, asymptotic behavior and infinite time blow up of solution for E(0) < d. In Section 4, we prove the global existence, asymptotic behavior and infinite blow up of solution for E(0) = d. In Section 5, we prove infinite time blow up of solutions for E(0) > 0 (ω = 0).

Preliminaries
In this subsection we give some notations for function spaces and some preliminary lemmas.
(a) For any function u ∈ W 1,p 0 (Ω), we have the inequality The best constant B q,p depends only on Ω, n, p and q.
Global Well-posedness of Solutions 5 (b) For any u ∈ W 1,p 0 (Ω), p ≥ 1 and r ≥ 1, the inequality where the constant C depends on n, p, q and r, Now, we define the energy associated with problem (1.1) by Lemma 2.4. Let u be a solution of problem (1.1). Then, Proof. Multiplying the first equation of problem (1.1) by u t and integrating from 0 to t with respect to t, we obtain which yields (2.3) by a simple calculation.

Potential wells
In this subsection, we establish the corresponding method of potential wells which is related to the logarithmic nonlinear term u p−2 ln |u|. First, we define the functionals Then, it is obvious that Similar in [25,29], the potential well depth is defined as where N is the Nehari manifold, which is defined by Next, we define Lemma 2.5. For any u ∈ W 1,p 0 (Ω), u p = 0 and let f (λ) = J(λu). Then we have Proof. By definition of J, we have Since u p = 0, we have lim λ→0 f (λ) = 0, lim λ→+∞ f (λ) = −∞, and So, we have Global Well-posedness of Solutions 7 Now, we denote r the constant given by where L p is defined as in Lemma 2.1.
(3) This conclusion is similar to the proof of (2).
Lemma 2.7. The following statements hold: (2) d has a positive lower bound, namely, where L p is defined as in Lemma 2.1.
(3) There is a positive function u ∈ N such that J(u) = d.
Proof. The proof of this lemma is almost the same to that of Lemma 7 in [29]. So we omit it here.
The proofs of Lemmas 2.8-2.10 are similar to those in [7,21], we omit the details.
3. Global existence, asymptotic behavior, and blow up for E(0) < d In this section we prove the global existence, asymptotic behavior, and blow up of solutions for problem (1.1) with E(0) < d.
Proof. Let {w j } be the eigenfunctions of the Laplacian operator subject to the Dirichlet boundary condition: which is chosen as the complete orthogonal basis of W 1,p 0 (Ω) and it is also a basis of H 1 0 (Ω). Construct the approximate solution u m (x, t) of problem (1.1) as and Multiplying (3.2) by g jm and summing for j, we get By integrating (3.5) with respect to t over (0, t), we have Noting that u m (0) → u 0 in W 1,p 0 (Ω) as m → ∞, we see that u 0 ∈ W implies u m (0) ∈ W for sufficiently large m. Coupled with (3.7) and an argument similar to the proof of Lemma 2.8, we have u m (t) ∈ W for sufficiently large m and 0 ≤ t < ∞. Hence (3.7) turns to  By taking µ < p, we deduce from Lemma 2.8 that On the other hand, by a direct calculation, we have where Ω 1 = {x ∈ Ω : |u m | < 1}, Ω 2 = {x ∈ Ω : |u m | ≥ 1}, ν is chosen such that p + ν ≤ p * as p < n and ν is positive as p ≥ n. By using Lemma 2.3, it follows from (3.10) that At this point, we choose ν > 0 such that θ(p + ν) < p, we deduce that Ω |u m | p ln |u m | dx ≤ Cd.
Integrating (3.2) with respect to t, we get (3.12) Therefore, up to a subsequence, by (3.9) and (3.11) we may pass to the limit in (3.12) and obtain a weak solution u(x, t) of problem (1.1) with the corresponding regularity stated above. On the other hand, from (3.3) and (3.4) we have u(x, 0) = u 0 (x) in W 1,p 0 (Ω) and u t (x, 0) = u 1 (x) in L 2 (Ω).

Asymptotic behavior for E(0) < d
where ε is a positive constant to be specified later. In order to show our result, we need the following lemma.
Lemma 3.3. There exist two positive constants α 1 and α 2 such that (3.14) Proof. It is easy to see that F (t) and E(t) are equivalent. So we omit it here.
Proof of Theorem 3.2. Taking a derivative of (3.13) with respect to t, using (1.1) and Lemma 2.4, we have It follows from (2.2), for a constant N > 0, we see that For N < p, by applying logarithmic Sobolev inequality (Lemma 2.1), we have Noting that ln u p p ≤ ln(p 2 J(t)) ≤ ln(p 2 E(t)) ≤ ln(p 2 E(0)) < ln(p 2 d).
For any fixed N , we choose ε sufficiently small satisfying Then inequality (3.15) becomes (3.16) Taking µ satisfying nL p pe e pβ/n < µ < p, Consequently, inequality (3.16) becomes From (3.14), we have A simple integration of (3.17) leads to Again This completes the proof. Proof. Arguing by contradiction, we suppose that the solution u is global. Then for any T > 0, we define the following auxiliary function It is clear that θ(t) > 0 for all t ∈ [0, T ]. In view of the continuity of θ(t), we obtain that there is a κ > 0 such that where κ is independent of T . Taking a derivative of (3.18) with respect to t, using (1.1) and (2.4), we obtain  It follows from (3.18)-(3.21), we see that In what follows, we will estimate the second term in (3.22). By using Hölder and Young inequalities, we obtain (3.25) (3.26) Inserting (3.26) into (3.22), we obtain that where On the other hand, from Lemmas 2.4, 2.9, (2.6), E(0) < d, and p > 2, we conclude that Then, by using (3.27), we can see clearly that Let Y (t) = ln(θ(t)), then and Therefore, the function Y (t) is increasing with respect to t. A simple integration of (3.28) leads to which implies that Namely  Proof. The proof of this lemma is almost the same to that of Lemma 4.1 in [32]. So, we omit the details. 4.1. Global existence for E(0) = d Theorem 4.2. Let u 0 ∈ W 1,p 0 (Ω) and u 1 ∈ L 2 (Ω) hold. Assume that E(0) = d and u 0 ∈ W , then problem (1.1) admits a global weak solution u(t) ∈ L ∞ (0, +∞; W 1,p 0 (Ω)) with u t (t) ∈ L ∞ (0, +∞; L 2 (Ω)).
The remainder proof is similar to that in Theorem 3.4.

Blow up for E(0) > 0
In this section, we state and prove the blow-up result of solution to of problem (1.1) with E(0) > 0 and ω = 0.
Lemma 5.3. Let u 0 ∈ W 1,p 0 (Ω) and u 1 ∈ L 2 (Ω) and u be a weak solution to problem (1.1) with the maximum existence time interval [0, T ). Assume that the conditions of Theorem 5.1 hold, then the solution u(t) belongs to V for problem (1.1).
Remark 5.4. The proofs of Lemmas 5.2 and 5.3 are similar to those in [21,32], we omit the details.