Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term

Abstract: The main goal of this work is to investigate the initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic term at three di erent initial energy levels, i.e., subcritical energy E(0) < d, critical initial energy E(0) = d and the arbitrary high initial energy E(0) > 0 (ω = 0). Firstly, we prove the local existence of weak solution by using contraction mapping principle. And in the framework of potential well, we show the global existence, energy decay and, unlike the power type nonlinearity, in nite time blow up of the solution with sub-critical initial energy. Then we parallelly extend all the conclusions for the subcritical case to the critical case by scaling technique. Besides, a high energy in nite time blow up result is established.


Introduction and main results
In this paper, we study initial boundary value problem of nonlinear wave equation with weak and strong damping terms and logarithmic source term u tt − ∆u − ω∆u t + µu t = u ln |u|, (x, t) ∈ Ω × ( , ∞), (1.1) u(x, t) = , x ∈ ∂Ω, t ≥ , (1.2) u(x, ) = u (x), u t (x, ) = u (x), x ∈ Ω, (1.3) where Ω ⊂ R n (n ≥ ) is a bounded domain with a smooth boundary ∂Ω, ω ≥ , µ > −ωλ , (1.4) λ being the rst eigenvalue of the operator −∆ under homogeneous Dirichlet boundary conditions. The undamped hyperbolic equation (1.5) was introduced by D'Alembert [1] to model the propagation of waves along vibrating elastic string. By introducing the potential well, the global existence and nite time blow up of solution to (1.5) with E( ) < d were proved by Payne and Sattinger in [2], [3] respectively. The nonlinear wave equation with linear weak damping term was considered by Levine and Serrin [4] in abstract form Pu tt + A(u) + Q(t, u t ) = F(u) (1.6) and they proved the nonexistence of global solution for the negative initial energy, i.e., E( ) < . Later, Pucci and Serrin [5] extended this results to E( ) < D , where D is positive. Vitillaro [6] studied the similar problem replacing the linear weak damping term by the nonlinear one u tt − ∆u + b|u t | m− u t = c|u| p− u (1.7) and derived the conditions of initial data leading to nite time blow up of the solution for E( ) ≤ E . For the initial boundary value problem of classical strongly damped wave equation Webb [7] gave the local existence uniqueness, global existence and the asymptotic behavior of the solution.
For the initial boundary value problem of strongly damped semilinear wave equation Pata and Squassina [8] proved the existence of the universal attractor, in the presence of a quite general nonlinearity of critical growth. Moreover, they obtained the asymptotic behavior of the solutions in dependence of the damping coe cient.
For the wave equation with both linear weak and strong damping terms, we can directly go to [9] for the most recent progress. Gazzola and Squassina [9] proved that the initial boundary value problem of the weak and strong damping hyperbolic equation (1.10) has a unique local solution, and the global existence and nonexistence results were also proved for E( ) ≤ d. Also, the nite time blow up of solution with high energy E( ) > d (ω = , µ > ) was obtained. The logarithmic nonlinearity is of much interest in physics, since it appears naturally in in ation cosmology and super symmetric led theories, quantum mechanics and nuclear physics [10], [11]. Haraux and Cazenave [12] proved the existence and uniqueness of solution for the Cauchy problem for the nonlinear Schrödinger equation and for the nonlinear Klein-Gordon equation Górka [13] obtained that the initial boundary value problem of logarithmic Klein-Gordon equation The weak damping with logarithmic wave equation was introduced by Hiramatsu [14] to model the dynamics of Q-ball in theoretical physics. Then its initial boundary value problem was considered by Han in [15] (1.15) and the global existence of weak solutions was proved, for all (u , u ) ∈ H × L in R . By constructing an appropriate Lyapunov function, Zhang and Liu [16] obtained the exponential decay estimates of energy with E( ) < d for all (u , u ) ∈ H × L , I(u ) > . Al-Gharabli and Messaoudi [17] proved the global existence and the exponential decay of solutions of the following plate equation for all (u , u ) ∈ H (Ω) × L (Ω) and < E( ) < d, I(u ) > . Later, Al-Gharabli and Messaoudi [18] considered a general damping h(u t ) instead of a linear u t one considered in [17], where h is a function having a polynomial growth near the origin. They established the global existence and the general decay of solutions for all (u , u ) ∈ H (Ω) × L (Ω), < E( ) < d and I(u ) > . As shown in the previous works, the dynamical behaviors of solutions are quite di erent when with presence of di erent nonlinearities, i.e., power type and logarithmic type. To be speci c, with the presence of logarithmic term, the nite time blow up of solution don't occur anymore which means the Nehari manifold in the initial energy space H no longer plays a role as a threshold separating global and non-global existence of solution. And instead in this article the results show that Nehari manifold can be viewed as a threshold which indicates the decay or in nite time blow up of solutions. In order to investigate and describe the dynamical behavior of solution that strongly relied on the initial data, we focus on the logarithmic term in three initial data levels, i.e., subcritical energy E( ) < d, critical initial energy E( ) = d and the arbitrary high initial energy E( ) > (ω = ).
The present paper is organized as follows. Section 2 presents some notations and preliminaries. In section 3, we state our main results. Section 4 prove the local existence of solution. Section 5 prove the global existence, asymptotic behavior and in nite time blow up of solution for E( ) < d. In section 6, the global existence, asymptotic behavior and blowup of solution for E( ) = d is proved. At last, in Section 7, we prove the in nite time blowup result for E( ) > (ω = ).

Notations and primary lemmas
In this section, we present some preliminaries to prove the main results. We denote the inner product and the norm on H (Ω) by (·, ·) and ∇ · , respectively. The symbol · will indicate the norm on L (Ω). Moreover, we denote by ·, · the duality pairing between H − (Ω) and H (Ω). For any v, w ∈ H (Ω), we have By (1.4), · * is an equivalent eccentric module over H (Ω).
First, for problem (1.1)-(1.3) we introduce the energy functional By I(u) we de ne the potential well (stable set) the outer space of potential well (unstable set) and the Nehari manifold The depth of potential well is de ned as On the other hand, as the di erence between two types of nonlinearities, assumptions on the power type nonlinearity don't work on the logarithmic one. Consequently, introducing the logarithmic Sobolev inequality and revisiting the corresponding estimates is a necessity to handle logarithmic nonlinear term u ln |u|. The following logarithmic Sobolev inequality was introduced by [20], Chapter 8.14 (also see [19] for a di erent proof). Lemma 2.1. [19,20] If u ∈ H (R n ) and a > . Then For u ∈ H (Ω), we can de ne u(x) = for x ∈ R n \Ω. Then u ∈ H (R n ), that is to say, for a general domain Ω, we have following logarithmic Sobolev inequality, where u ∈ H (Ω) and a > .
(I) This conclusion is similar to the proof of (ii). Proof. From the de nition of d in (2.8) and Lemma 2.3 in (iii) , it follows that u ∈ N. As a result, we obtain (2.16) By virtue of (2.4), I(u) = and (2.16), we obtain which gives (2.14).

Lemma 2.5. Let u be a solution of problem (1.1)-(1.3), then E(t) is a non-increasing function with respect to t.
Proof. Multiplying Eq. (1.1) by u t and integrating it over Ω × [s, t), we can obtain Thus, the proof is completed.

Main results
In this section, we state our main results about problem (1.1)-(1.3).
there exist two positive constantsĈ and ξ independent of t such that    Arguing by contradiction, we suppose that there exist two solutions v and w such that (4.3) hold. Then by subtracting the obtained two equations and testing with v t − w t , we can derive which directly says w ≡ v. So we complete the proof. Again Lemma 4.1 simpli es that for ∀ u ∈ U T , there exists v = Φ(u) such that v is the unique solution to problem (4.3). Next we prove that for a suitable T > , Φ is a contractive map satisfying Φ(U T ) ⊂ U T .
First we can conclude that where v = Φ(u) is the corresponding solution, to problem (4.3) for xed u ∈ U T . Similar to the arguments of (4.6), we can derive that Hence, (4.7) becomes Therefore, taking η = v t in (4.10) and arguing as above, gives for some δ < as long as T is su ciently small. So by the Contraction Mapping Principle, we can conclude that problem (1.1)-(1.3) admits a unique solution.

Sub-critical initial energy . Global existence for E( ) < d.
We namely, Arguing by contradiction, we suppose that there exists the rst t ∈ ( , T) such that I(u(t )) = and I(u(t)) > for ≤ t < t , i.e., From the de nition of N, then we have u(t ) ∈ N, which implies that J(u(t )) ≥ d. From (2.8) and the de nition of E(t), it holds that which contradicts (5.3). Then u(t) ∈ W for all t ∈ [ , T), which together with (5.4) and (2.17) gives Thus, the proof of Lemma 5.1 is completed.
which together with the Logarithmic Sobolev inequality leads to Here and in the sequel we denote by C i > , i = , , as constants.
we get Hence, from inequality (5.8) it follows that where C > is a constant independent of t. Therefore similar as the proof of Lemma 4.1 we know that problem (1.1)-(1.3) admits a global weak solution.

. Asymptotic behavior for E( ) < d.
Next, we can now prove the asymptotic behavior of the solution to problem (1.1)-(1.3), which relies on the construction of a Lyapunov functional by performing a suitable modi cation of the energy.

Proof of Theorem 3.3
First, we de ne where ε > will be chosen later. From Lemma 5.1 and (2.1)-(2.3) it implies that It is easy to see that L(t) and E(t) are equivalent in the sense that there exist two positive constants β > and β > depending on ε such that Taking the derivative of L(t) with respect to time yields together with Eq. (1.1). Now, we estimate the last term on the right hand side of (5.12) as follows. By using Young's inequality, we obtain, Substituting (5.13) into (5.12) with (2.1) gives that By logarithmic Sobolev inequality, we have Recalling (2.1), (2.4) and E(t) ≤ E( ) < d, we get ln u < ln( J(u)) < ln d. (5.16) Now, choosing M < , and ε small enough such that Then inequality (5.15) becomes and then we have Further, by virtue of (5.11), let ξ = Mε β , (5.20) becomes Therefore, we get By Schwarz inequality, we can get These three inequalities entail η(t) ≥ for any t ∈ [ , T]. As a consequence, we reach the following di erential inequality which means G(t)G (t) − (G (t)) > . On the other hand, by directly calculation, we can see that By (5.36), we know that (ln G(t)) is increasing with respect to t, using this fact, integrating (5.35) from t to t, we have where ≤ t ≤ t. Then Since G( ) = and G ( ) > , we can take t su ciently small such that G (t ) > and G(t ) > . Then for su ciently large t, i.e., Proof. Arguing by contradiction we suppose that t uτ * dτ ≡ for ≤ t < T, which gives u t ≡ for ≤ t < T. So we can get u(t) ≡ u (x), which contradicts the assumption of Lemma 6.1.
. Global existence for critical initial energy. Proof. We prove that u(t) ∈ W for ≤ t < T. Arguing by contradiction we suppose that there exists a t ∈ ( , T) such that I(u(t )) = and u(t ) ≠ , which says J(u(t ) ≥ d. Hence, by we get t uτ * dτ = and u t = for ≤ t ≤ t , which implies du dt = and u(x, t) = u (x) for x ∈ Ω, ≤ t ≤ t . Thus we can conclude I(u(t )) = I(u ) > , which contradicts I(u(t )) = .
Now we turn to prove the existence of global solution to problem (1.1)-(1.3) with E( ) = d.
Proof of Theorem 3.5 First Theorem 3.1 gives the existence of the local solution. From Lemma 6.1 we can get (6.1). By (2.18) and E( ) = d we obtain Moreover, from Lemma 6.2 it follows u(t ) ∈ W. Let v(t) = u(t + t ) and t ≥ , then v(t) is a solution of problem (1.1)-(1.3), which combining Theorem 3.2 says that T = +∞.

. Asymptotic behavior for E( ) = d.
Next, we can now prove the asymptotic behavior of the solution to problem (1. Proof of Theorem 3.6 From Theorem 3.5 it implies that there exists a t > such that E(t ) < d, I(u(t )) > or u(t ) = , which together with Theorem 3.3 says, whereĈ * =Ĉe −ξt .
. In nite time blow up for the critical initial energy. By the similar arguments as those in the proof of Theorem 3.5, together with Lemma 6.1 and Lemma 6.3. we can set E(t ) < d and I(u(t )) < . The remainder of the proof is the same as Theorem 3.4.

In nite time blow up for E( ) > (ω = )
We rst prove the following lemmas to obtain that the unstable set V is invariant with respect to t under the ow of problem (1.1)-(1.3) with E( ) > , ω = and µ ≥ .
Proof. We prove u(t) ∈ V for t ∈ [ , T). Arguing by contradiction, we suppose that t ∈ ( , T) is the rst time such that I(u(t )) = (7.8) and I(u(t) < for t ∈ [ , t ).
Hence from Lemma 7.2 it follows that the map is strictly increasing on the interval [ , t ), which together with (7.2) gives that This tells us that lim t→+∞ G(t) = +∞.
The remainder of the proof is similar to that of Theorem 3.4.