BLOW-UP OF POSITIVE-INITIAL-ENERGY SOLUTIONS FOR NONLINEARLY DAMPED SEMILINEAR WAVE EQUATIONS

. We consider a class of semilinear wave equations with both strongly and nonlinear weakly damped terms, u tt − ∆ u − ω ∆ u t + µ | u t | m − 2 u t = | u | p − 2 u, associated with initial and Dirichlet boundary conditions. Under certain con- ditions, we show that any solution with arbitrarily high positive initial energy blows up in finite time if m < p . Furthermore, we obtain a lower bound for the blow-up time.


Introduction
In this contribution, we study the blow-up of solutions of the following initial boundary value problem of a semilinear wave equation: x ∈ ∂Ω, t > 0, u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ Ω. (1.1) Here, Ω is a bounded domain of R n with a smooth boundary ∂Ω. Additionally, we assume that u 0 ∈ H 1 0 (Ω), u 1 ∈ L 2 (Ω), (1.2) and ω, µ, m and p are positive constants, with 2 < p ≤ 2n n−2 , for n ≥ 3, 2 < p < ∞, for n = 2. (1. 3) The linear strong damping term −ω∆u t appears in models describing Kelvin-Voigt materials that exhibit both elastic and viscous properties, while the nonlinear frictional damping term µ|u t | m−2 u t usually models external friction forces such as air resistance acting on the vibrating structures.

MOHAMED AMINE KERKER
In the absence of strong damping (ω = 0), the equation in (1.1) reduces to the nonlinearly damped wave equation (1.4) Eq. (1.4) was first studied by Levine [8] in the case of linear weak damping (m = 2). By using the concavity method, he showed that solutions with negative initial energy blow up in finite time. Later, for the case m > 2, by using a different method, Georgiev and Todorova [4] [1,6,5,9,12,14,15,18] and the references therein.
In related work, Messaoudi [11] considered He proved, for m < p, that solutions with E(0) < d blow up in finite time. Later, by using a modified concavity method, Kafini and Messaoudi [7] established a blow-up result for (1.5) when the damping term is linear. When m > 2, by introducing a new technique, Song [13] obtained a finite time blow-up result for solutions of (1.5) with arbitrarily high initial energy. In this paper, motivated by the above-cited works, we give sufficient conditions for the finite time blow-up of solutions of (1.1) in both cases: E(0) < 0 and E(0) > 0. Furthermore, we give a lower bound for the blow-up time.

Preliminaries
We denote by ∥ · ∥ p the L p (Ω) norm (2 ≤ p < ∞), and by (·, ·) the L 2 inner product. The notation ⟨·, ·⟩ is used in this paper to denote the duality paring between H −1 (Ω) and H 1 0 (Ω). We introduce the energy functional By simple calculation we have

Blow-up with negative initial energy
In this section we show that the solution of (1.1) blows up in finite time if m < p and E(0) < 0. To prove the main result in this section, we define H(t) := −E(t) and we use the following lemma. For the proof, see [10]. Proof. To obtain a contradiction, we suppose that the solution of (1.1) is global; then, for every fixed T > 0, there exists a constant K > 0 such that We now define an auxiliary function for ε small (to be chosen later) and By taking the derivative of L(t) we obtain To estimate the last term in the right-hand side of (3.5), we use Young's inequality By taking for a large constant k to be chosen later, (3.6) becomes Combining (3.5) and (3.7), and using (3.3), yields Therefore, in view of the last inequality, (3.8) becomes where Writing p = (p + 2)/2 + (p − 2)/2 in (3.10) yields where We choose now k large enough such that the coefficients γ i , for 2 ≤ i ≤ 4, are positive. Once k is fixed, we choose ε small enough such that γ 1 > 0 and L(0) > 0.
Hence, the inequality (3.11) becomes for some constant A > 0. Consequently, we have Next, by using Hölder's and Young's inequalities, we obtain for 1/s + 1/r = 1. We take s = 2(1 − α), which gives s Therefore, by using Lemma 3.1, we obtain From (3.1) and (3.2), we have (3.14) So, by using Jensen's inequality, we get and by combining it with (3.13) and (3.14), we deduce From the inequalities (3.12) and (3.15), we finally obtain the differential inequality for some D > 0. A simple integration of (3.16) over (0, t) immediately yields (3.17) which shows that the functional L(t) blows up in finite time. □ Remark 3.3. From (3.17) we obtain the following upper bound of the blow-up time:

Blow-up with positive initial energy
In this section, we consider the blow-up of solutions of the problem (1.1) when E(0) > 0. To prove the main theorem of this paper, we employ the following lemma.
Proof. By the convexity of the function u x /x for u ≥ 0 and x > 0. □ for some M > 0 to be specified later in the proof, then u(t) blows up in finite time.

Lower bound for the blow-up time
In this section, we give a lower bound for the blow-up time T max . To this end, we define where β, A 1 and A 2 are positive constants to be determined later in the proof.