Global existence and blow-up of solutions for logarithmic Klein-Gordon equation

Abstract: This arcitle concerns the initial-boundary value problem for a class of Klein-Gordon equation with logarithmic nonlinearity. By using Galerkin method and compactness criterion, we prove the existence of global solutions to this problem. Meanwhile, the blow-up of solutions in the unstable set is also obtained.

As m = 0, for sufficiently large initial data, the blow-up results of the problem (1.7)-(1.9) in finite time was proved by H. A. Levine [14] and J. Ball [15]. Furthermore, Y. C. Liu [16], L. E. Payne and D. H. Sattinger [17] and D. H. Sattinger [18] obtained the results of the global existence and nonexistence of weak solutions for the problem (1.7)-(1.9) by establishing the method of potential wells. Also in [16,19], the authors gave a threshold result of solutions and obtained the vacuum isolating of solutions.
At last we should mention that the logarithmic heat equation was studied by H. Chen and S. Y. Tian [20] and H. Chen, P. Luo and G. W. Liu [21]. Moreover, there were also many researches on the logarithmic Schrödinger equation [22][23][24][25].
In this paper, by applying Galerkin method and compactness criterion, we prove the global existence of the problem (1.1)-(1.3). Furthermore, in the sense of L 2 norm, the blow-up result for this problem is obtained by the concavity method.

Some lemmas
For the applications through this paper, we list up some known lemmas.
Then the function u is called a weak solution of (1.1)-(1.3) on [0, T ].
Lemma 2.5 [28] Let u n (x) be a bounded sequence in L p (Ω), 1 ≤ p < +∞ such that u n almost everywhere converges to u. Then u ∈ L p (Ω) and u n weakly converges in L p (Ω) to u, where Ω ⊂ R n is a bounded domain.

Potential wells
At first, we introduce some useful functionals for u ∈ H 1 0 (Ω).
As in [17], the potential well depth is defined as Now, we define the Nehari manifold ( [29,30]) by The stable set W and the unstable set U can be defined respectively by It is to see that the potential well depth d may also be described as (2.7) Lemma 2.6 Let u ∈ H 1 0 (Ω) and u 0, then we have Proof. (i) By lim λ→0 + λ 2 log λ = 0, lim λ→+∞ log λ = +∞ and (ii) By direct calculations, we obtain Let d dλ J(λu) = 0, then we deduce that From (3.2), we have (2.10) By (2.9) and (2.10), the equality (2.8) is valid. (2.11) Proof. By Lemma 2.2, we have (2.12) By taking a = √ 2π, we obtain from (2.12) that Combining Lemma 2.6 and (2.5) yields that (2.14) We receive from (2.13) and Lemma 2.6 that In order to further study the problem (1.1)-(1.3), for 0 < ε < 1 and u ∈ H 1 0 (Ω), we define some functionals as follows where λ 1 is the first eigenvalue of the following boundary value problem (2.22) Proof. By u ∈ N ε (u), we get From (2.17) and Lemma 2.2 that (2.24) By taking a 2 = 2πε in (2.24), we obtain which implies that u 2 ≥ a n e n+3 .
Proof. (a) is easy to be proved. Here we omit the proof of it.

Global existence of solutions
In this section, by applying Galerkin method and the compactness principle, we study the global solutions of the problem (1. j=1 be a basis for H 1 0 (Ω). We are going to find out the approximate solution u m (t) in the form u m (t) = m j=1 g jm (t)ω j with g jm (t) ∈ C 2 [0, T ], ∀T > 0, where the unknown functions g jm (t) are determined by the following ordinary differential equation By the density of H 1 0 (Ω) in L 2 (Ω), there exist α jm and β jm , j = 1, 2, · · · , m such that By a Picard's iteration method, there exists solution g jm (t) of the problem (3.1) and (3.2) in interval [0, t 1 m ) for some t 1 m ≤ T . From the uniformly boundedness of function g jm (t) and the extension theorem, we can extend this solution to the whole interval [0, T ] for any given T > 0 by making use of the a priori estimates below.
The proof of Theorem 3.1 is completed.
Proof. By using the similar argument as Theorem 3.1, we are going to prove Theorem 3.3. Under the conditions in Theorem 3.3, by Lemma 3.1, we have u 0 ∈ W ε for ε ∈ (ε 1 , ε 2 ). For any given ε 1 < ε < ε 2 , we derive J ε (u m (0)) > 0 and E m (0) < d(ε), which implies that u m (0) ∈ W ε . Once again, we get u m (t) ∈ W ε by Lemma 3.1. Here, the approximate solutions u m (t) are given in the proof of Theorem 3.1.
Multiplying both sides of (3.1) by g jm (t), summing over j from 1 to m and integrating with respect to t, we obtain The remainder of the proof for Theorem 3.3 is the same as those of Theorem 3.1. Here, we omit them.

Blow-up of solution
In this section, we establish the blow-up property of solution for the problem (1.1)-(1.3).
Proof. It follows from Lemma 2.3 that By contradiction, we assume that there exists t * ∈ [0, +∞) such that u(t * ) U, then, from the continuity of K(u(t)) on t, we have K(u(t * )) = 0. This implies that u(t * ) ∈ N. We get from (2.7) that J(u(t * )) ≥ d, which is contradiction with (4.1). Consequently, Lemma 4.1 is valid.

Conclusions
By applying logarithmic Sobolev inequality, the Galerkin method and compactness theorem, we prove the global existence results of the problem (1.1)-(1.3) under the conditions that the initial values u 0 ∈ W, u 1 ∈ L 2 (Ω) satisfy (i) 0 < E(0) < d or (ii) K(u 0 ) ≥ 0 and E(0) = d. Meanwhile, under the condition of positive initial energy, by using the concavity analysis method, we establish the finite time blow-up result of solutions in the sense of L 2 norm. On the other hand, the global existence of solution for this problem is also obtained in a family of potential wells W ε . Our result implies that the polynomial nonlinearity is important for the solutions of such kinds of Klein-Gordon equation to be blow-up in finite time.