NON-EXISTENCE OF GLOBAL SOLUTIONS TO NONLINEAR WAVE EQUATIONS WITH POSITIVE INITIAL ENERGY

. We consider the Cauchy problem for nonlinear abstract wave equations in a Hilbert space. Our main goal is to show that this problem has solutions with arbitrary positive initial energy that blow up in a ﬁnite time. The main theorem is proved by employing a result on growth of solutions of abstract nonlinear wave equation and the concavity method. A number of examples of nonlinear wave equations are given. A result on blow up of solutions with arbitrary positive initial energy to the initial boundary value problem for the wave equation under nonlinear boundary conditions is also obtained.

1. Introduction. The paper is devoted to the question of nonexistence of global solutions of second order abstract wave equations of the form P u tt + Au + Qu t = F (u), (1) under the initial conditions u(0) = u 0 , u t (0) = u 1 (2) in a real Hilbert space H with the inner product (·, ·) and the corresponding norm · . Here A is densely defined selfadjoint positive definite operator in a Hilbert space H, P is selfadjoint densely defined positive operator and Q is selfadjoint densely defined non-negative operator in H so that ) → H is a nonlinear gradient operator with the potential G(·) which satisfies the condition for some α > 0, R 0 ≥ 0. As far as we know first result about nonexistence of a global solution of an evolution equation of the form (1) in a Hilbert space H is a global non-existence theorem obtained by using the energy method in [34] for the equation (1) with P = I, Q = 0 and a nonlinear gradient operator F (·) with potential G(u) that satisfy the conditions (F (u), u) ≥ νG(u), (F (u), u) ≥ G( u 2 ), ∀u ∈ D(A where ν > 2 is a given number and a continuous function G : R + → R + satisfies the condition < ∞ for some v 0 > 0. One of the effective techniques which has been employed to demonstrate when a solution to a nonlinear partial differential equation blows up in a finite time is the concavity method. The idea of the concavity method introduced in [20] is based on a construction of some positive functional Ψ(t) = ψ(u(t)), which is defined in terms of the local solution of the problem (the local solvability of the problem is therefore required) and proving that the function Ψ(t) satisfies the inequality (5) given in the following statement: 20]). Let Ψ(t) be a positive, twice differentiable function, which satisfies the inequality for some α > 0. If Ψ(t 0 ) > 0 and Ψ (t 0 ) > 0, then there exists a time t 1 ≤ t 0 + Ψ(t0) The concavity method and its modifications were used in the study of various nonlinear partial differential equations (see e.g. [2,11,12,14], [21]- [24], [15,28,30,32,33]).
One of the main conditions on initial data guaranteing nonexistence of a global solution of problems considered in these papers is negativity of the initial energy of the corresponding problem. In a number of papers employing the potential well theory introduced in [29] it is shown that some solutions of nonlinear wave equations with positive energy may blow up in a finite time.
In recent years considerable attention has been given to the question of global non-existence of solutions to initial boundary value problems for nonlinear wave equations with arbitrary positive initial energy.
The concavity method and its modifications are employed to find sufficient conditions of blow up of solutions with arbitrary large positive initial energy to the Cauchy problem and initial boundary value problems for nonlinear Klein -Gordon equation, damped Kirchhoff-type equation, generalized Boussinesq equaton, quasilinear strongly damped wave equations and some other equations (see, e.g. [1,4], [16]- [18], [25,26,35,36] and references therein).
Our main goal is to show that non-existence of global solutions with arbitrary positive initial energy of the problem (1) actually can be established for wider class of nonlinear wave equations than equations considered in the preceeding papers by using the Lemma 1.1 and a modification of the following theorem on growth of solutions of the problem (1), (2) with Q = 0.
We would also like to note that our main result on nonexistence of global solutions of the problem (1), (2) is new even for the case of undamped equations, i.e. equations of the form (1) with Q ≡ 0.  -EXISTENCE OF GLOBAL SOLUTIONS TO NONLINEAR WAVE EQUATIONS   989 for some α > 0, u is a solution of the problem (1), (2) and the initial data satisfy the conditions and Then, lim t→+∞ (u(t), P u(t)) = +∞, if u(·) exists on (0, +∞).
2. Non-existence of global solutions to abstract wave equation. In this section we show that a version of the Theorem 1.2 due to [20] is valid also for the solutions of the damped equation (1). We show that if u(t) is a global solution of the problem (1),(2) for initial data satisfying some conditions (including initial data with arbitrary positive initial energy), then it becomes unbounded as t → ∞. Then by using this result we prove our main Theorem 2.2 about nonexistence of global solutions of the problem (1), (2).
Employing the equation (1) in (15), utilizing the condition (4) and the definition of the energy (12) we obtain Thanks to the energy identity (11) we deduce from this inequality the following one: First suppose that the condition (8) holds true. Then, the last inequality implies that Therefore Ψ(t) → +∞ and Ψ (t) → +∞ as t → +∞. Suppose now that (9) holds true. Then, upon multiplying the inequality (16) by Ψ(t) ≥ 0, in view of the equality (14), we may write In view of the definition (13) of Ψ(t), by an application of the Cauchy-Schwarz inequality for the inner product we obtain that, the difference composed of the first two terms on the right hand side of (17) is nonnegative. Consequently, we deduce that where Following [20] we define the positive functional H(t) := Ψ −α (t) and multiplying the inequality (18) by −αΨ −2−α (t) we see that H(t) satisfies Now, suppose that which translates in terms of H as We claim that H (t) < 0 for all t ≥ 0. If not, then there is a smallest time δ > 0 such that Hence, upon multiplying the inequality (20) by H (t) we deduce the inequality Integrating this inequality and rearranging we get Thanks to the condition (22) the right hand side of the inequality (23) is strictly positive. Hence, the left hand side must be strictly positive for each t ∈ [0, δ]. This contradicts our assumption H (δ) = 0. Thus, we conclude that This, in turn, implies that the inequality (23) is valid for all t ≥ 0, and the left hand side of the inequality (23) is strictly positive for all t ≥ 0. Also note that, as H (t) < 0, the first term in brackets on the left hand side of the inequality (23) is strictly negative. Thus, we conclude that the second term in brackets is also strictly negative for all times, that is we have Translating back this inequality equivalently reads This inequality implies that, Ψ(t) grows as t 2 , and Ψ (t) grows as t, as time increases. Let us note that, only the right hand side of the last inequality involves the positive parameter C 0 . Therefore, we choose so that the right hand side of (21) is minimal. It is easy to check that, with this choice of C 0 , the condition (21) is equivalent to the second condition in (9). The first statement of the theorem, that is easily follows from the inequality (24). This finishes the proof.
Next, suppose that u(t) is a global solution to the problem (1), (2) with initial data [u 0 , u 1 ], that satisfy the condition (8). From the inequality (16) we immediately deduce that This implies the statement of the theorem.
Theorem 2.2. Suppose that the operators P, Q, A and F satisfy all the conditions of Theorem 2.1. Let u(t) be the solution to the problem (1)-(2) with initial data satisfying one of conditions (8) or (9). Further suppose that, there exists a 0 , a 1 > 0 such that Then, there exists t 1 > 0 such that Moreover, there are infinitely many initial data [u 0 , u 1 ] with arbitrary large initial energy E(u 0 , u 1 ) for which the corresponding solutions blow up in a finite time, i.e.  For local existence of weak and strong solutions of a Cauchy problem for second order evolution equations of the form (1), we refer, e.g. to [3][5] [13] and references therein. Let us note that under some natural restrictions on the nonlinearity and its potential, local solvability of the problem (1),(2) can be established by Galerkin method (see e.g. [5]).
Hence, we deduce from the previous inequality that, there exists t * 0 ≥ 0 such that, for all t ≥ t * 0 we have which is the main inequality in the assumptions of the Lemma 1.1. Moreover, as stated by Theorem 2.1, we also have that Ψ (t) → +∞ as t → +∞, which implies that we can find t 0 ≥ t * 0 such that Ψ (t 0 ) > 0. Finally, both of the assumptions (8) and (9) on the initial data necessarily imply that u 0 ≡ 0, which ensures by our choice (25) of C 0 that, Ψ(t) ≥ C 0 > 0 for any t ≥ 0, in particular Ψ(t 0 ) > 0. Hence, we see that all assumptions of the Lemma 1.1 are satisfied after the time t = t 0 , and consequently Ψ(t) → +∞ in a finite time. This contradicts to our initial assumption that the solution was global. It remains to show that the last statement of the theorem holds true. Suppose that u 0 is an arbitrary element of D such that Note that, the last condition is necessary to have the condition (9).
The condition (7) for this choice of u 1 takes the form which can be seen to be equivalent to In view of (29) this inequality is satisfied for µ 2 small enough. Moreover, the initial energy has the form and it is clear that E(u 0 , u 1 ) can take arbitrary large values for η large enough.
Finally, let us note that the condition (29) is satisfied, for instance, if the potential G(·) satisfies the conditions Clearly (29) is satisfied for u 0 = λw with λ 1 and w ∈ D such that G(w) > 0.

Remark 1.
It is also worth mentioning that, the virtue of the conditions (9) of the Theorem 2.2 lies in the fact that, it provides blowing up solutions with initial data having arbitrary large positive initial energies. For small positive initial energies however, this condition can not recapture what is already known, for instance, for the problem where Ω ⊂ R 3 is a bounded domain with smooth boundary ∂Ω. The potential well theory gives a full characterization of the behaviour of the solutions in terms of global existence/nonexistence for initial energies less than the depth of the potential well. For example, take R 0 = 0 and suppose that the initial data verify E(0) = 0. In this case we know from the potential well theory that, all nonzero solutions must blow up in finite time (see [29]), whereas the conditions (9) only contains initial data that satisfy the condition (u 0 , u 1 ) > 0.
3. Examples of nonlinear wave equations.

Example 1. Nonlinear Klein-Gordon equation.
Let u be a local strong solution to the Cauchy problem where m > 0 is a given number, u 0 ∈ H 1 (R 3 ), u 1 ∈ L 2 (R 3 ) are given compactly supported functions. The equation (31) can be written in the form (1) with P = I, A = −∆ x + m 2 I and F (u) = |u| 2 u. It is easy to see that, this nonlinearity satisfies the condition (4) with α = 1 2 > 0 and R 0 = 0. Moreover, the condition (26) holds with a 0 = m 2 . Thus, it follows from Theorem 2.2 that, if the initial energy is nonnegative, and where Ω ⊂ R n is a bounded domain with smooth boundary ∂Ω, p is an arbitrary positive number if n = 1, 2 and p ∈ (0, 2 n−2 ] if n ≥ 3. Let us note that this result easily follows from the results of T. Cazenave obtained in [8] for solutions of the problem (33) and the Theorem 1.2 of H. A. Levine.
Indeed, T. Cazenave proved that each solution of the problem (33) either blows up in a finite time or is uniformly bounded.
Thus, if the functions u 0 , u 1 satisfy the conditions of Theorem 1.2, that is then the corresponding local solution of the problem (33) can not be continued on the whole interval [0, ∞), i.e. it must blow up in a finite time. For results on local solvability of the Cauchy problem and initial boundary value problems for semilinear Klein -Gordon equations see, e.g., [7].

Example 2. Generalized Boussinesq equation.
Similarly we can find sufficient conditions for blow up of solutions with arbitrary positive initial energy for the generalized Boussinesq equation under the homogeneous Dirichlet boundary conditions where f (u) = |u| m u + P m−1 (u), m ≥ 1 is a given integer, ν > 0 is a given number, Ω ∈ R n is a bounded domain and P m−1 (u) is a polynomial of order ≤ m − 1. Since Ω is bounded, the Poincaré inequality assures that the assumption (26) is verified. Hence, the conclusion of Theorem 2.2 holds provided that the assumptions of Theorem 1.2 are fulfilled, that is u 0 , u 1 satisfy Example 3. Nonlinear plate equations. It is clear that we can apply Theorem 2.2 to find sufficient conditions of blow up of solutions to initial boundary value problems for the nonlinear plate equations of the form where f (·) : R → R is a continuous function which satisfies the condition where  • Cauchy problem and initial boundary value for system of nonlinear Klein-Gordon equation where h 1 , h 2 ∈ L 2 (R 3 ) are given functions, m, µ, b, β are positive numbers. • Initial boundary value problem under homogeneous Dirichlet boundary condition for the strongly damped wave equation where P is some polynomial of order less than p + 1.
• Initial boundary value problem under homogeneous Dirichlet boundary condition for nonlinear wave equation with structural damping term where f : R → R is a continuous function that satisfies the condition (35). For local solvability of initial boundary value problem for equations (36) and (37) we refer to [6], where the authors employed the fact that the semigroups generated by corresponding linear problems are analytic (see [9]). • Initial boundary value problem for quasilinear strongly damped wave equation of the form where p > m ≥ 2 are given numbers. This problem can be written as an abstract Cauchy problem (1), (2) in the Hilbert space H = L 2 (Ω) with Since p > m ≥ 2, the condition (4) is satisfied with α = p 4 − 1 2 > 0 and R 0 = 0. • Initial boundary value problem for nonlinear Love equation: Here f (u) = |u| p−2 u + P (u) + h(x), P (u) is a polynomial of order < p − 1, h ∈ L 2 (0, 1) is a given function, m > 2, p > 2, b > 0, c > 0, a > 0 ∈ R, are given numbers. We assume also that m < p when a > 0. This problem can be written as a Cauchy problem (1), (2) in the Hilbert space H = L 2 (0, 1) with Employing Young's inequality we can show that the condition (4) is satisfied with some α > 0 and R 0 > 0. Local solvability of the problem (38) as well as blow up of solutions of this problem with nonpositive initial energy when a < 0, f (u) = |u| m−2 u is discussed in [27].
(43) Employing (43), similar to the proof of the Theorem 1.2 we can show that if (u 0 , u 1 ) u 0 2 > 2E(u 0 , u 1 ) + then Ψ(t) = u(t) 2 → +∞ as t → +∞. Finally arguing as in the proof of the Theorem 2.2 we get the inequality Thanks to the last inequality and the Lemma 1.1 we proved the following  where Γ 1 ∪Γ 2 = ∂Ω, mes(Γ 1 ) = 0 and the nonlinear term f (·) satisfies the condition (35). Notice that the first result on blow up of solutions of the equation (39) under nonlinear boundary conditions (41) with negative initial energy was obtained in [22], in [24] it is shown that there are solutions with positive initial energy that blow up in a finite time.