In this paper, we consider the following quasilinear wave equation with time delay and acoustic boundary conditions:

Here, J = [0, T), 0 < T ≤ ∞, a: Ω → R⁺ is a positive function, l, α, β ≥ 2, μ₁ > 0, μ₂ is a real number, and τ > 0 represents the time delay. Further, Ω is a regular and bounded domain of Rⁿ(n ≥ 1) and ∂Ω(:= Γ) = Γ₀ ∪ Γ₁, where Γ₀ and Γ₁ are closed and disjoint and ∂∂ν denotes the outer normal derivative. The functions k, q, h: Γ₁ → R⁺(:= [0,∞]) are essentially bounded and 0 < q₀ ≤ q(x) on Γ₁.

The acoustic boundary conditions were introduced by Morse and Ingard [16] and developed by Beale and Rosencrans in [1], where the authors proved the global existence and regularity of the linear problem. Other authors have studied the existence and decay of solutions for a viscoelastic wave equation with acoustic boundary conditions (see [3, 4, 6, 7, 12, 13, 15, 19, 20, 23] and the references therein).

The time delay arises in many physical, chemical, biological and economical phenomena because these phenomena depend not only on the present state but also on the past history of the system in a more complicated way. In particular, the effects of time delay strikes on our system have a significant effect on the range of existence and the stability of the system. The differential equations with time delay effects have become an active area of research, see for example [9, 11, 17, 18]. In [14], without the delay term and the acoustic boundary condition, Liu and Wang considered the global nonexistence of solutions with the positive initial energy for a class of wave equations:

where J = [0, T), 0 < T ≤ ∞, Ω is a bounded regular open subset of Rⁿ(n ≥ 1), l, α, β ≥ 2 and a, Q, f satisfy some conditions. Recently, for l = 2, a(x) = 1, Q(u_t) = a|u_t|^m−2u_t, μ₁ = μ₂ = 0, f(u) = b|u|^p−2u, and without the time delay term in our system, Jeong at al [8] investigated the global nonexistence of solutions for a quasilinear wave equation with acoustic boundary conditions

where a, b > 0, α, β, m, p > 2 are constants and Ω is a regular and bounded domain of Rⁿ(n ≥ 1) and ∂Ω(= Γ) = Γ₀ ∪ Γ₁. Here Γ₀ and Γ₁ are closed and disjoint. The functions h, f, q: Γ₁ → R⁺ are essentially bounded. Moreover, for a(x) = 1, l = 2, div(|∇u_t|^β−2∇u_t) = 0, Q = 0, and without boundary conditions, Kafini and Messaoudi [10] studied the following nonlinear damped wave equation

where p > m ≥ 2, b, μ₁ are positive constants, μ₂ is a real number, and τ > 0 represents the time delay. They proved the blow-up result in a nonlinear wave equation with time delay and without acoustic boundary conditions.

Motivated by the previous works, we consider an equation in a broader and more generalized form than the system discussed above. So we study the global nonexistence of solutions for a quasilinear wave equation with the time delay and acoustic boundary conditions. To the best of our knowledge. there are no results of a quasilinear wave equations with the time delay and acoustic boundary conditions. Thus the result in this work is very meaningful. The main result will be proved in Section 3.

In this section, we shall give some notations, assumptions and a theorem which will be used throughout this paper. We denote by m′ the Hölder conjugate of m, ||u||_p = ||u||_L^p(Ω), ||u||_p,Γ = ||u||_L^p(Γ), ||u||_1,s = ||u||_W^1,s(Ω), where L^p(Ω) and W^1,s(Ω) stand for the Lebesgue spaces and the classical Sobolev spaces, respectively. Specially we introduce the set

We make the following same assumptions on a, Q, f as section 4.2 of [22].

(H1)a(x) ∈ L^∞(Ω) such that a(x) ≥ a₀ a.e. in Ω for some a₀ > 0.

(H2)f(x, u) ∈ C(Ω×ℝⁿ,ℝⁿ) and f(x, u) = ∇_uΦ(x, u), with normalizing condition Φ(x, 0) = 0.

There are constants d₁ > 0, p > α and μ < μ₀a₀ such that

for all x ∈ Ω and u ∈ ℝⁿ. Moreover, there is ε₁ > 0 such that for all ε ∈ (0, ε₁] there exists d₂ = d₂(ε) > (p – α)d₁/p such that

for all x ∈ Ω.

(H3) There are m > 1 and a measurable function d = d(x, t) defined on Ω×J such that d(·, t) ∈ L^p/(p−m)(Ω) for a.e. t ∈ J and

for all values of the arguments x, t, v, where

Remark 2.1

We note that when Q(x, t, u_t) = b(1+t)^ρ|u_t|^m−2u_t, −∞ < ρ ≤ m−1, condition (H3) holds.

Now, we transform the equation (1.1)–(1.7) to the system, using the idea of [21] and introduce the associated energy. So, we introduce the new variable:

Thus, we have

Then problem (1.1)–(1.7) takes the following form:

We introduce the following space

for some T > 0.

We state, without a proof, a local existence which can be established by combining arguments of [2, 5, 24].

Theorem 2.1

Letu0∈WΓ01,α(Ω), u₁ ∈ L²(Ω), f₀ ∈ L²(Ω×(0, 1)) and y₀ ∈ L²(Γ₁) be given. Suppose that l, α, β, m, p > 2, max{l, β, m} < α < p < nα/(n – α), μ₁ > |μ₂| and (H1)–(H3) hold. Then problem(2.6)–(2.13) has a unique local solution (u, z, y) ∈ Z×L²([0, T);L²(Ω×(0, 1)))×L²([0, T);L²(Γ₁)) for some T > 0.

In order to state and prove our result, we introduce the energy functional

where

We set

where B₁ is the best constant of the Sobolev embedding W01,α(Ω)↪Lp(Ω) given by

We also set

In this section, we state and prove our main result. Our main result as follows.

Theorem 3.1

Letu0∈WΓ01,α(Ω), u₁ ∈ L²(Ω), f₀ ∈ L²(Ω×(0, 1)) and y₀ ∈ L²(Γ₁) be given. Suppose that l, α, β, m, p > 2, max{l, β, m} < α < p < nα/(n – α), μ₁ > |μ₂| and (H1)–(H3) hold. Assume further that

Then the solution (u, z, y) ∈ Z × L²(R₊);L²(Ω × (0, 1))) × L²(R₊);L²(Γ₁)) of problem (2.6)–(2.13) can not exist for all time.

In this section, we shall prove Theorem 3.1. We start with a series of lemmas. We denote

Theorem 3.1 will be proved by contradiction, so we shall suppose that the solution of (2.6)–(2.13) exists on the whole interval [0,∞), i.e. T = ∞.

Lemma 3.1

Let (u, z, y) be the solution of (2.6)–(2.13). Then the energy functional defined by (2.15) satisfies, for some constant c₀ > 0,

Lemma 3.2

If (λ₀, E(0)) ∈ ∑, then we have

Lemma 3.3

For all t ∈ J, we have