BLOW UP OF SOLUTIONS FOR A SEMILINEAR HYPERBOLIC EQUATION

u(x, 0) = u0(x), ut(x, 0) = u1(x), x ∈ Ω, (1.3) where α > 0, β > 0 and u0, u1 are given functions. A is a second order elliptic operator where the coefficients are depended on x and t. f and g are some functions specified later. In the case A = −∆, many mathematicians studied the problem (1.1)−(1.3). For α = 0, g(v) ≡ 0 (absence of the damping term), the source term f(u), in the case where the initial energy is negative, causes the blow up of solutions (see [1, 8]). In contract, in the absence of the source term (β = 0), the damping term (with α = 0) assures global existence for arbitrary initial data (see [7, 9]). The interaction between the damping and the source terms was considered by Levine [9, 10] in linear damping case (α = 0, g(v) ∼= v) and polynomial source term of the form f(u) = |u|u, p > 2. He showed that the solutions with negative initial energy blow up in finite time. Georgiev and Todorova [5] extended Levine’s result to the nonlinear case, where the damping term is given by |ut| ut, m > 2. Precisely, they showed that the solution continues to exist globally ’in time’ if m ≥ p and blows up in finite time if m < p and the initial energy is sufficiently negative. Vitillaro [16]


Introduction
Let Ω be a bounded domain of R n with a smooth boundary ∂Ω.We are concerned with the blow up of solutions of an initial-boundary value problem for a semilinear hyperbolic equation with dissipative terms: u(x, 0) = u 0 (x), u t (x, 0) = u 1 (x), x ∈ Ω, ( where α > 0, β > 0 and u 0 , u 1 are given functions.A is a second order elliptic operator where the coefficients are depended on x and t. f and g are some functions specified later. In the case A = −∆, many mathematicians studied the problem (1.1)− (1.3).For α = 0, g(v) ≡ 0 (absence of the damping term), the source term f (u), in the case where the initial energy is negative, causes the blow up of solutions (see [1,8]).In contract, in the absence of the source term (β = 0), the damping term (with α = 0) assures global existence for arbitrary initial data (see [7,9]).The interaction between the damping and the source terms was considered by Levine [9,10] in linear damping case (α = 0, g(v) ∼ = v) and polynomial source term of the form f (u) = |u| p−2 u, p > 2.He showed that the solutions with negative initial energy blow up in finite time.Georgiev and Todorova [5] extended Levine's result to the nonlinear case, where the damping term is given by |u t | m−2 u t , m > 2. Precisely, they showed that the solution continues to exist globally 'in time' if m ≥ p and blows up in finite time if m < p and the initial energy is sufficiently negative.Vitillaro [16] extended the result in to situation when the damping is nonlinear and the solution has positive initial energy.Recently, Yu [17] studied the same problem of Vittilaro with strongly damping term.He proved that the solution exists globally if E(t) < d, m < p and blows up in finite time in unstable set.G.Li and al [11] considered the Petrovsky equation and proved the global existence of the solution under conditions without any relation between m and p, and established an exponential decay rate.They also showed that the solution blows up in finite time if p > m and the initial energy is less than the potential well depth.
Messaoudi in [14] studied the following problem: where a, b > 0, p, m > 2.He showed that if the initial energy is negative, then the solutions blow up in finite time.
In this work, we will prove that if the initial energy is positive, then the solution of problem (1.1) − (1.3) blows up in finite time.

preliminaries
In this section we shall give some assumptions and notations which will be used throughout this work.H 1 ) The elliptic operator A is defined as follows: where ≤ n is symmetric and there exists a constant a 0 > 0 such that : We assume that the function g(v) is increasing and g(v) ∈ C 0 (R) ∩ EJQTDE, 2012 No. 40, p. 2 C 1 (R * ).Furthermore, there exist two positive constants k 0 and k 1 such that: for all v ∈ R and 2 < m < ∞.
H 3 ) The function f ∈ C 0 (R, R + ), with the primitive where s ∈ R, c 0 > 0 and p > 2. A typical example of these functions is Next we introduce some notations, which will be used in the sequel: where L r (Ω) is the Lebesgue space.
Remark.By using Poincaré's inequality and the Sobolev embedding theorem.Then, there exists a constant C * depending on Ω, r only such that

Local existence of solutions
To allow for studying the local existence and blow up of solutions, we proceed to obtain a variational formulation of the problem (1.1) − (1.3).By multiplying equation (1.1) by v ∈ H 1 0 (Ω), integrating over Ω and using integration par parts, it is easy to verify that under the hypothesis (H 1 ) the problem (1.1) − (1.3) is equivalent to the following variational problem: By using the hypothesis (H 1 ), we verify that the bilinear form a(., .): which implies that a(., .) is coercive.
Referring to [3] and [5], by using the precedent hypotheses we can demonstrate the following theorem, which confirms the local existence and uniqueness of a weak solution.
, then there exists T > 0 such that the problem (1.1) − (1.3) has a unique local solution u(t) having the following regularities :

Blow-up of solutions
In this section, we will establish our main blow-up result concerning the problem (1.1) − (1.3).We set We define the energy function associated to the solution u of the problem (1.1) − (1.3) by By using arguments similar to those used by Vitillaro [16], we prove the following Lemma, which is very important to obtain the blow-up result.
Then any solution of (1. From (4.9), (4.2) and (H 3 b), we get For ε small to be chosen later, we then define the following auxiliary function: Let us remark that G is a small perturbation of the energy.By taking the time derivation of (4.12) and using a variational formulation, we obtain that By using (4.2), (H 3 ) and (4.9) from (4.14) we deduce that : Using the assumption (H 2 b), we get Then we exploit the following Young's inequality : with r = m and s = m m−1 to get for all positive constant δ.By using Holder's inequality and (2.1) we get where c(λ), c 1 (λ) are positive constants.Inserting (4.16), (4.17) and (3.2) in (4.15), we arrive at We observe that where λ 1 is given in Lemma 4.2.From (4.5), it follows: where , using Lemma 4.2, we have C 1 > 0 and by (4.9), we see that At this point we choose δ so that δ − m−1 m = MH −σ (t), for M a large constant to be determined later, and substituting in the last inequality, we obtain Since p > m, we have , where C 3 is a positive constant depending on Ω only.We also have from (4.11) Exploiting the following algebraic inequality: At this point we choose λ > 0, (it is the case where ) such that and we can choose M > Once M is fixed, we pick ε small enough such that Then, from (4.23) we deduce that: where  Remark.For E(0) < 0, we set H(t) = −E(t), instead of (4.9) and use arguments similar to those used in the proof of Theorem 4.3 to deduce that the solution blows up in finite time.