ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO A QUASILINEAR

HYPERBOLIC EQUATION WITH NONLINEAR DAMPING

Mohammed Aassila

Institut de Recherche Mathématique Avancée, Université Louis Pasteur et C.N.R.S., 7 rue René Descartes, 67084 Strasbourg Cédex, France. E-mail: aassila@math.u-strasbg.fr

Abstract: We prove the existence and uniqueness of a global solution of a damped quasilinear hyperbolic equation. Key point to our proof is the use of the Yosida approximation. Furthermore, we apply a method based on a specific integral inequality to prove that the solution decays exponentially to zero when the time t goes to infinity.

Key words and phrases: nonlinear damping, global existence, Yosida approximation, integral inequality, exponential decay.

AMS Subject Classification: 35B40, 35L70.

1. Introduction

Let Ω ⊂ ℝ^{N} be a bounded domain with smooth boundary Γ. In this paper we are concerned with the global existence and asymptotic behavior of solutions to the mixed problem

(P)  \{ \begin{matrix}u′′ − f(∥∇u∥_{2}^{2})Δu + g(u′) = h(x, t) in Ω × ℝ_{+}, \\ u = 0 on Γ_{0} × ℝ_{+}, \\ \frac{∂u}{∂ν} = 0 on Γ_{1} × ℝ_{+}, \\ u(x, 0) = u_{0}(x), u′(x, 0) = u_{1}(x) in Ω. \end{matrix}

Here f(·) is a C^{1}-class function satisfying f(s) ≥ m_{0} > 0 for s ≥ 0 with m_{0} constant, {Γ_{0}, Γ_{1}} is a partition of Γ such that Γ̄_{0} ∩ Γ̄_{1} = ∅, Γ_{0} ≠ ∅, Γ_{1} ≠ ∅, and g is a continuous increasing odd function such that g′(x) ≥ τ > 0.

Physically, the problem (P) occurs in the study of vibrations of flexible structures in a bounded domain. The motivation for incorporating internal material damping in the quasilinear wave equation as in the first equation of (P) arises from the fact that inherent small material damping is present in real materials. Hence from the physical point of view we say that internal structural damping force will appear so long as the system vibrates.

The problem (P) with h = 0, g(x) = δx (δ > 0) and Γ_{1} = ∅ was studied by De Brito [3]. She has shown the existence and uniqueness of global solutions for sufficiently small initial data by using a Galerkin method. When g(x) = δx and Γ_{0} = ∅, Ikehata [4] has shown the existence of global solutions by a Galerkin method, the key point of his proof is to restrict (P) to the range of −Δ on which −Δ is positive definite. In fact the restricted problem can be solved by a Galerkin method exactly as in De Brito [3]. When g is nonlinear, Ikehata’s approach seems to be very difficult. The author in [1] has been successful in proving the global existence and establishing the precise decay rate of solutions when Γ_{1} = ∅, g is nonlinear without any smallness conditions on the initial data and without the assumption g′(x) ≥ τ > 0.

Our study is motivated by Ikehata and Okazawa’s work [5] where global existence was proved when g(x) = δx (δ > 0) and Dirichlet or Neumann boundary condition by using the Yosida approximation method together with compactness arguments. In our work, the feedback g is nonlinear, and furthermore we study the asymptotic behavior of the global solution when h = 0.

The contents of this paper are as follows. In section 2, we give our main results. In section 3, we establish the existence of global solutions. In section 4, we study the asymptotic behavior of solutions of (P) with h = 0.

2. Statement of the main theorems

We define the energy of the solution u to problem (P) by the formula

(2.1)  E(t) = \frac{1}{2}∥u′(t)∥_{2}^{2} + f̄(∥∇u∥_{2}^{2})

where f̄ = ∫_{0}^{s} f(t) dt and ∥ · ∥_{n} denotes the usual norm of L^{n}(Ω). Our main results are

Theorem 2.1

For any (u_{0}, u_{1}) ∈ (H^{2}(Ω)∩H_{Γ_{0}}^{1}(Ω))×H_{Γ_{0}}^{1}(Ω) and h ∈ L^{1}(0, ∞; H^{1}(Ω))∩ L^{∞}(0, ∞; L^{2}(Ω)) satisfying

(2.2)  \frac{B_{1}C_{1}}{m_{0}} [(|Δu_{0}|^{2} + \frac{1}{m_{0}} |∇u_{1}|^{2})^{\frac{1}{2}} + \frac{1}{\sqrt{m_{0}}} ∫_{0}^{∞} |∇h| dt] < τ

there exists a unique solution u(t) on [0, ∞) to problem (P) such that

u ∈ L^{∞}(0, ∞; H^{2}(Ω) ∩ H_{Γ_{0}}^{1}(Ω)) ∩ BC([0, ∞), H_{Γ_{0}}^{1}(Ω)),

u′ ∈ L^{∞}(0, ∞; H_{Γ_{0}}^{1}(Ω)) ∩ BC([0, ∞), L^{2}(Ω)),
u′′ ∈ L^{∞}(0, ∞; L^{2}(Ω)),
u′′ − f(∥∇u∥_{2}^{2})Δu + g(u′) = h in L^{2}(Ω) a.e. on (0, ∞)
u = 0 on Γ_{0} × ℝ_{+},
\frac{∂u}{∂ν} = 0 on Γ_{1} × ℝ_{+},
u(x, 0) = u_{0}(x), u′(x, 0) = u_{1}(x) in Ω.

Here B_{1}, C_{1} are positive constants defined by

(2.3)  C_{1} = \sqrt{2E(0)} + ∫_{0}^{∞} |h| dt

B_{1} = max_{0≤s≤ \frac{C^{2}_{1}}{m_{0}}} |f′(s)|

and BC([0, ∞); L^{2}(Ω)) denotes the set of all L^{2}(Ω)- bounded continuous functions on [0, ∞).

Theorem 2.2

In addition to the conditions in theorem 2.1, we assume that f is nondecreasing, h = 0 and

(2.5)  |g(x)| ≤ C_{2}|x| if |x| ≤ 1,
(2.6)  |g(x)| ≤ 1 + C_{3}|x|^{q} if |x| > 1 (q ≥ 1)

then we have the decay property E(t) ≤ E(0)e^{1−t/C_{4}} for all t ≥ 0

where C_{4} = C(Ω)(1 + E(0)^{\frac{q−1}{2q}} .

Before giving the proofs, we recall the: Lemma 2.3 ([5] Lemma 3.1)

Let A be a nonnegative selfadjoint operator in a Hilbert space H with the norm | · |, A_{λ} its Yosida approximation and (A_{λ})^{\frac{1}{2}} the square root of A_{λ} (λ > 0). Then

(2.8)  ∥(A_{λ})^{\frac{1}{2}} ∥ ≤ \frac{1}{\sqrt{λ}} (λ > 0),

(2.9)  |v − J \frac{1}{2} ; λ v| ≤ \sqrt{λ}|(A_{λ})^{\frac{1}{2}} v|, v ∈ H,

EJQTDE, 1998 No. 7, p. 3 here J_{λ} = (I + λA)^{−1} (λ > 0). Lemma 2.4 ([6] Theorem 8.1)

Let E : ℝ_{+} → ℝ_{+} be a non-increasing function and assume that there exists a constant T > 0 such that

∫_{t}^{∞} E(s) ds ≤ T E(t) ∀t ∈ ℝ_{+},

then

E(t) ≤ E(0)e^{1− \frac{t}{T}} ∀t ≥ T.

Lemma 2.5 ([5] Lemma 3.2) Let F and G be nonnegative continuous functions on [0, T]. If

F (t)^{2} ≤ C + ∫_{0}^{t} F (s)G(s) ds on [0, T],

then

F (t) ≤ \sqrt{C} + \frac{1}{2} ∫_{0}^{t} G(s) ds on [0, T],

where C > 0 is a constant.

Lemma 2.5 is a special case of an inequality that can be found in Bihari [2].

3. Global existence

Let −Δ_{λ} (λ > 0) be the Yosida approximation of −Δ and (∇_{λ})^{1/2} be the square root of −Δ_{λ}, that is (∇_{λ})^{1/2} = (−Δ_{λ})^{1/2}(I + λΔ)^{−1/2}.

First, we solve the approximate problem

(P_{λ})  \{ \begin{matrix}u′′_{λ} − f(∥∇_{λ}u_{λ}∥_{2}^{2})Δ_{λ}u_{λ} + g(u′_{λ}) = h in Ω × ℝ_{+}, \\ u_{λ}(0) = u_{0} ∈ H^{2}(Ω) ∩ H_{Γ_{0}}^{1}(Ω), u′_{λ}(0) = u_{1} ∈ H_{Γ_{0}}^{1}(Ω). \end{matrix}

Problem (P_{λ}) can be easily solved by successive approximation method. Hence problem (P_{λ}) has a unique local solution u_{λ} ∈ C^{1}([0, T_{λ}), L^{2}(Ω)) on some interval [0, T_{λ}). We shall see that u_{λ}(t) can be extended to [0, ∞). Lemma 3.1

Let C_{1} be the constant defined by (2.3). Then the following inequality holds

(3.1)  |u′_{λ}|^{2} + m_{0}|∇_{λ}u_{λ}|^{2} + 2τ ∫_{0}^{t} |u′_{λ}|^{2} ds ≤ C_{1}^{2}.

Proof Multiplying both sides of the first equation in (P_{λ}) by 2u′_{λ}, we have

\frac{d}{dt}|u′_{λ}|^{2} + f(|∇_{λ}u_{λ}|^{2})\frac{d}{dt}|∇_{λ}u_{λ}|^{2} + 2(g(u′_{λ}), u′_{λ}) = 2(h, u′_{λ}) a.e. on [0, T_{λ}).

After integration on [0, t], we see that

|u′_{λ}(t)|^{2} + f̄(|∇_{λ}u_{λ}|^{2}) + 2τ ∫_{0}^{t} |u′_{λ}(s)|^{2} ds ≤ 2E(0) + 2 ∫_{0}^{t} |h(s)||u′_{λ}(s)| ds.

It follows from lemma 2.5 that

|u′_{λ}|^{2} + m_{0}|∇_{λ}u_{λ}|^{2} + 2τ ∫_{0}^{t} |u′_{λ}(s)|^{2} ds ≤ (\sqrt{2E(0)} + ∫_{0}^{∞} |h(s)| ds)^{2} := C_{1}^{2}.

Lemma 3.2

Set

Z_{λ}(t) = |Δ_{λ}u_{λ}(t)|^{2} + \frac{|∇_{λ}u′_{λ}(t)|^{2}}{f(|∇_{λ}u_{λ}(t)|^{2})}.

Assume that on [0, T_{λ})

(3.3)  |\frac{d}{dt}f(|∇_{λ}u_{λ}(t)|^{2})| ≤ 2τf(|∇_{λ}u_{λ}(t)|^{2})

then for t ∈ [0, T_{λ}) we have

(3.4)  Z_{λ}(t)^{1/2} ≤ (|Δu_{0}|^{2} + \frac{1}{m_{0}} |∇u_{1}|^{2})^{1/2} + \frac{1}{\sqrt{m_{0}}} ∫_{0}^{∞} |∇h(s)| ds.

Proof

Multiplying the both sides of the first equation in (P_{λ}) by −2Δ_{λ}u′_{λ}(t), we have

\frac{d}{dt}|∇_{λ}u′_{λ}(t)|^{2}+f(|∇_{λ}u_{λ}(t)|^{2})\frac{d}{dt}|Δ_{λ}u_{λ}(t)|^{2} = 2(h, −Δ_{λ}u′_{λ})−2g′(u′_{lambda})|∇_{λ}u′_{λ}(t)|^{2}.

It follows that

f(|∇_{λ}u_{λ}(t)|^{2})Z′_{λ}(t) ≤ 2|∇_{λ}h||∇_{λ}u′_{λ}(t)| − [ \frac{\frac{d}{dt}f(|∇_{λ}u_{λ}|^{2})}{f(|∇_{λ}u_{λ}|^{2})} + 2τ] |∇_{λ}u′_{λ}(t)|^{2}.

By (3.3) we obtain

Z′_{λ}(t) ≤ \frac{2}{\sqrt{m_{0}}} |∇_{λ}h|Z_{λ}(t)^{1/2}.

Integrating this inequality on [0, t], we have

Z_{λ}(t) ≤ Z_{λ}(0) + \frac{2}{\sqrt{m_{0}}} ∫_{0}^{t} |∇_{λ}h|Z_{λ}(s)^{1/2} ds.

Since Z_{λ}(0) ≤ |Δu_{0}| + \frac{1}{m_{0}} |∇u_{1}|^{2}, it follows from lemma 2.5 that

Z_{λ}(t)^{1/2} ≤ (|Δu_{0}|^{2} + \frac{1}{m_{0}} |∇u_{1}|^{2})^{1/2} + \frac{1}{\sqrt{m_{0}}} ∫_{0}^{t} |∇_{λ}h| ds,

then we obtain (3.4). Lemma 3.3

Set

α_{λ}(t) = \frac{1}{f(|∇_{λ}u_{λ}|^{2})} |\frac{d}{dt}f(|∇_{λ}u_{λ}|^{2})| .

If (2.2) is satisfied, then we have

α_{λ}(t) < 2τ on [0, T_{λ}).

Proof

First we show that

α_{λ}(0) < 2τ.

Since

|\frac{d}{dt}f(|∇_{λ}u_{λ}|^{2})| ≤ 2B_{1}|u′_{λ}||Δ_{λ}u_{λ}| ≤ 2B_{1}C_{1}Z_{λ}(t)^{1/2},

we have by definition

(3.8)  α_{λ}(t) ≤ 2B_{1}C_{1} \frac{Z_{λ}(t)^{1/2}}{f(|∇_{λ}u_{λ}|^{2})} ≤ \frac{2}{m_{0}} B_{1}C_{1}Z_{λ}(t)^{1/2}.

Setting t = 0 in (3.7), we see from (2.2) that α_{λ}(0) < 2τ. Now suppose that (3.6) does not hold on [0, T_{λ}). Since α_{λ}(t) is continuous, (3.7) implies that there is a t^{∗} > 0 such that α_{λ}(t) < 2τ on [0, t^{∗}), and

(3.9)  α_{λ}(t^{∗}) = 2τ

i.e. (3.3) is satisfied on [0, t^{∗}]. Therefore, it follows from lemma 3.2 and (2.2) that

(3.10)  Z_{λ}(t^{∗})^{1/2} < \frac{m_{0}}{B_{1}C_{1}} τ.

Combining (3.10) with (3.8), we obtain α_{λ}(t^{∗}) < 2τ. This contradicts (3.9).

EJQTDE, 1998 No. 7, p. 6 Lemma 3.4

Set

C_{5} = (|Δu_{0}|^{2} + \frac{1}{m_{0}} |∇u_{1}|^{2})^{1/2} + \frac{1}{\sqrt{m_{0}}} ∫_{0}^{∞} |∇h|.

If (2.2) is satisfied, B_{1}C_{1}C_{5} < m_{0}τ, then

|Δ_{λ}u_{λ}(t)|^{2} + \frac{1}{B_{0}} |∇_{λ}u′_{λ}(t)|^{2} ≤ C_{5}^{2} on [0, T_{λ}),

where we set B_{0} = max_{0≤s≤ \frac{C^{2}_{1}}{m_{0}}} M(s), and hence

(3.12)  |u′′_{λ}(t)| ≤ B_{0}C_{5} + C_{6} + ess sup{h(s) : 0 ≤ s < ∞}

where C_{6} = max_{0≤x≤C_{1}} |g(x)|.

Proof

It follows from lemmas 3.2 and 3.3 that Z_{λ}(t) ≤ C_{5}^{2}, so we obtain (3.11). Next, multiplying the both sides of the first equation in (P_{λ}) by u′′_{λ}(t), we have

|u′′_{λ}(t)|^{2} = (h − g(u′_{λ}) − f(|∇_{λ}u_{λ}|^{2})Δ_{λ}u_{λ}, u′′_{λ}) a.e. on (0, T_{λ}).

Therefore, (3.12) follows from lemma 3.1 and lemma 3.4.

Lemma 3.5

Assume that (2.2) is satisfied. Then for any λ > 0 there exists a unique global solution u_{λ} ∈ C^{1}([0, ∞), L^{2}(Ω)) of the approximate problem (P_{λ}) such that u′_{λ}(·) is locally absolutely continuous on [0, ∞) and the first equation in (P_{λ}) holds a.e. on [0, ∞).

Proof

Let u_{λ}(t) be a solution of (P_{λ}) on [0, T_{λ}). Since u′_{λ}(t) and u′′_{λ}(t) are uniformly bounded in L^{2}(Ω), u_{λ}(T_{λ}) and u′_{λ}(T_{λ}) exist and we can choose them as new initial values. Moreover, since u_{λ}(t) is uniformly bounded, the local Lipschitz continuity of the mapping u → f(|∇_{λ}u|^{2})Δ_{λ}u is always verified. Therefore, u_{λ}(t) can be extended onto the semi-infinite interval [0, ∞).

Lemma 3.6

There is a subsequence {u_{λ_{n}}(·)} of {u_{λ}(·)} and u(·) ∈ BC([0, ∞), L^{2}(Ω)) such that for any T > 0

(3.13)  u_{λ_{n}}(·) → u(·) in C([0, T], L^{2}(Ω)) as n → ∞,

where λ_{n} > 0 (n ∈ ℕ) and λ_{n} → 0 (n → ∞), BC([0, ∞), L^{2}(Ω)) is the set of all L^{2}-valued bounded continuous functions on [0, ∞).

Proof

By the fact that ∥u_{λ}∥_{2} is bounded on [0, T_{λ}) and lemma 3.1, it follows that J_{λ}^{1/2}u_{λ} and ∇_{λ}u_{λ} belong to BC([0, ∞), L^{2}(Ω)). By the definition of ∇_{λ} we have

J_{λ}^{1/2}u_{λ}(t) = (I + ∇)^{−1}(J_{λ}^{1/2}u_{λ}(t) + ∇_{λ}u_{λ}(t)),

this implies that for each t > 0, {J_{λ}^{1/2}u_{λ}(t)} is bounded in H_{Γ_{0}}^{1}(Ω), and then relatively compact in L^{2}(Ω). As {J_{λ}^{1/2}u_{λ}(·)} is equicontinuous, we can apply the Ascoli-Arzela theorem to {J_{λ}^{1/2}u_{λ}(·)} in C([0, T], L^{2}) for any T > 0. Thus, there exist a subsequence {J_{λ_{n}}^{1/2}u_{λ_{n}}(·)} and u(·) ∈ BC([0, ∞), L^{2}) such that for any T > 0

(3.14)  J_{λ_{n}}^{1/2}u_{λ_{n}}(·) → u(·) in C([0, T], L^{2}) as n → ∞.

By (2.9) we conclude that for any T > 0

u_{λ_{n}}(·) → u(·) in C([0, T], L^{2}) as n → ∞.

Lemma 3.7

Let {λ_{n}} and u(·) be as in lemma 3.6. Assume that (2.2) is satisfied. Then u(·) ∈ BC^{1}([0, ∞), L^{2}) and there is a subsequence {μ_{n}} of {λ_{n}} such that for any T > 0

(3.15)  u′_{μ_{n}}(·) → u′(·) in C([0, T], L^{2}) as n → ∞.

Furthermore, u(·) ∈ L^{∞}(0, ∞; H^{2} ∩ H_{Γ_{0}}^{1}), u′(·) ∈ L^{∞}(0, ∞; H_{Γ_{0}}^{1}) and

(3.16)  Δ_{λ_{n}}u_{λ_{n}} → Δu weakly in L^{2} as n → ∞,

(3.17)  ∇_{μ_{n}}u′_{μ_{n}} → ∇u′ weakly in L^{2} as n → ∞.

Here BC^{1}([0, ∞), L^{2}) := {u ∈ BC([0, ∞), L^{2}); u′ ∈ BC([0, ∞), L^{2})}.

Proof

As J_{λ_{n}}^{1/2}u′_{λ_{n}} and ∇_{λ_{n}}u′_{λ_{n}} belong to BC([0, ∞), L^{2}) and that J_{λ_{n}}^{1/2}u′′_{λ_{n}} ∈ L^{∞}(0, T; L^{2}), (3.15) can be proved in the same way as in the proof of lemma 3.6, in fact we have

u(t) = u_{0} + ∫_{0}^{t} v(s) ds with v(s) = lim_{n→∞} u′_{μ_{n}}(s).

Since Δ_{λ_{n}}u_{λ_{n}} and ∇_{μ_{n}}u′_{μ_{n}} belong to BC([0, ∞), L^{2}), (3.16) and (3.17) follow from (3.13) and (3.15) respectively. Therfore we have

(3.18)  |Δu| ≤ lim inf_{n→∞} |Δ_{λ_{n}}u_{λ_{n}}| ≤ C_{5},
(3.19)  |∇u′| ≤ lim inf_{n→∞} |∇_{μ_{n}}u′_{μ_{n}}| ≤ \sqrt{B_{0}}C_{5}

i.e. u ∈ L^{∞}(0, ∞; H^{2} ∩ H_{Γ_{0}}^{1}) and u′ ∈ L^{∞}(0, ∞; H_{Γ_{0}}^{1}). Lemma 3.8

Let u and {λ_{n}} be as in lemma 3.6. Assume that (2.2) is satisfied. Then u ∈ BC([0, ∞); H_{Γ_{0}}^{1}) and for any T > 0

(3.20)  ∇_{λ_{n}}u_{λ_{n}} → ∇u in C([0, T], L^{2}) as n → ∞,

and hence

(3.21)  f(|∇u|^{2})Δu = weak lim_{n→∞} f(|∇_{λ_{n}}u_{λ_{n}}|^{2})Δ_{λ_{n}}u_{λ_{n}}.

Proof

We have

|∇_{λ_{n}}u_{λ_{n}} − ∇u|^{2} = |∇_{λ_{n}}u_{λ_{n}}|^{2} − |∇u|^{2} + 2(u − J_{λ_{n}}^{1/2}u_{λ_{n}}, −Δu). By (3.14) and (3.18)-(3.19) it suffices to show that

(3.22)  |∇_{λ_{n}}u_{λ_{n}}|^{2} → |∇u|^{2} in C([0, T]) as n → ∞,

which is equivalent to

(u_{λ_{n}}, −Δ_{λ_{n}}u_{λ_{n}}) → (u, −Δu) in C([0, T]) as n → ∞. But since (u, −Δu) − (u_{λ_{n}}, −Δ_{λ_{n}}u_{λ_{n}}) = (u − J_{λ_{n}}u_{λ_{n}}, −Δu) + (u_{λ_{n}}, −Δ_{λ_{n}}u + Δ_{λ_{n}}u_{λ_{n}})

= (u − J_{λ_{n}}u, −Δu) + (u − u_{λ_{n}}, −Δ_{λ_{n}}u) + (−Δ_{λ_{n}}u_{λ_{n}}, u − u_{λ_{n}}), we have |(u, −Δu)−(u_{λ_{n}}, −Δ_{λ_{n}}u_{λ_{n}})| ≤ λ_{n}|−Δu|^{2} +(|−Δu|+|−Δ_{λ_{n}}u_{λ_{n}}|)|u−u_{λ_{n}}|. Hence (3.22) follows from (3.13) and (3.18)-(3.19). Thus we obtain (3.20). Furthermore as

|∇u| = lim_{n→∞} |∇_{λ_{n}}u_{λ_{n}}| ≤ \frac{C_{1}}{\sqrt{m_{0}}} ,

EJQTDE, 1998 No. 7, p. 9 we see that u ∈ BC([0, ∞), H_{Γ_{0}}^{1}). For the proof of (3.21), since f(·) is of class C^{1}, we can use the mean value theorem, and so the proof follows from (3.1), (3.16) and (3.22).

Lemma 3.9

Let u and {μ_{n}} be as in lemma 3.7, then u′ has a (strong) derivative u′′ ∈ L^{∞}(0, ∞; L^{2}) and

(3.23)  u′′_{μ_{n}} → u′′ weakly in L^{2} as n → ∞ a.e.

and hence

(3.24)  u′′ − f(∥∇u∥_{2}^{2})Δu + g(u′) = h a.e.

Proof

¿From (3.12), we note that u′ is Lipschitz continuous. Therefore u′ is differentiable a.e. on (0, ∞) with u′′ ∈ L^{∞}(0, ∞; L^{2}). It follows from the previous lemma that

u′′_{μ_{n}} → w weakly in L^{2} (n → ∞),

where w = h − g(u′) + f(∥∇u∥_{2}^{2})Δu. So we see from the Banach-Steinhauss theorem that

∫_{t}^{t+h}(w(s), z) ds = lim_{n→∞} ∫_{t}^{t+h}(u′′_{μ_{n}}, z) ds z ∈ L^{2}.

It then follows from (3.15) that

\frac{1}{h} ∫_{t}^{t+h}(w(s), z) ds = \frac{(u′(t + h) − u′(t), z)}{h}.

Passing to the limit h → 0, we obtain w = u′′ a.e. on (0, ∞).

Lemma 3.10

Let u be as in lemma 3.6. Assume that (2.2) is satisfied, then u is the unique solution to problem (P).

The proof follows immediately from the Gronwall’s lemma.

4. Asymptotic behavior

In this section we consider the problem

(P 1)  \{ \begin{matrix}u′′ − f(∥∇u∥_{2}^{2})Δu + g(u′) = 0 in Ω × ℝ_{+}, \\ u = 0 on Γ_{0} × ℝ_{+}, \\ \frac{∂u}{∂ν} = 0 on Γ_{1} × ℝ_{+}, \\ u(x, 0) = u_{0}(x), u′(x, 0) = u_{1}(x) in Ω. \end{matrix}

The energy defined by (2.1) is such that

E′(t) = − ∫_{Ω} u′g(u′) dx ≤ 0,

hence the energy is non-increasing.

Multiplying the first equation in (P1) with u and integrating by parts, we obtain

(4.1) 2 ∫_{0}^{T} E(t) dt = − [∫_{Ω} uu′ dx]_{0}^{T} + 2 ∫_{0}^{T} ∫_{Ω}(u^{′2} − ug(u′)) dxdt+

+ ∫_{0}^{T} ∫_{Ω}(∫_{0}^{|∇u|^{2}} f(s) ds) dxdt − ∫_{0}^{T} ∫_{Ω} f(|∇u|^{2})|∇u|^{2} dxdt

for all 0 < T < +∞.

Whence, since f is non-decreasing, we obtain

(4.2)  2 ∫_{0}^{T} E(t) dt ≤ − [∫_{Ω} uu′ dx]_{0}^{T} + 2 ∫_{0}^{T} ∫_{Ω}(u^{′2} − ug(u′)) dxdt.

¿From now on, we shall denote by c(Ω) different positive constants which depend only on Ω. It is easy to verify that

(4.3)  − [∫_{Ω} uu′ dx]_{0}^{T} + 2 ∫_{0}^{T} ∫_{Ω} u^{′2} dxdt ≤ c(Ω)E(0).

By hypotheses (2.5)-(2.6) we have

(4.4)  |∫_{Ω} ug(u′) dx| ≤ c(Ω)E^{1/2}|E′|^{1/2} + c(Ω)E^{1/2}|E′|^{q/(q+1)}.

We apply the Young inequality to the two terms of the RHS of (4.4), we obtain

(4.5)  c(Ω)E^{1/2}|E′|^{1/2} ≤ c(Ω)|E′| + \frac{1}{3}E,

and

(4.6)  c(Ω)E^{1/2}|E′|^{q/(q+1)} = c(Ω)(|E′|^{\frac{q}{q+1}} E^{\frac{q−1}{2(q+1)}} )(E^{\frac{1}{q+1}} )

≤ c(Ω)E(0)^{\frac{q−1}{2q}} |E′| + \frac{1}{3}E.

Therfore, we conclude that

(4.7)  ∫_{0}^{T} E(t) dt ≤ c(Ω)(1 + E(0)^{\frac{q−1}{2q}})E(0),

that is

∫_{0}^{+∞} E(t) dt ≤ c(Ω)(1 + E(0)^{\frac{q−1}{2q}})E(0),

and by lemma 2.4 we arrive at

E(t) ≤ E(0)e^{1− \frac{t}{γ}} ∀t ≥ 0

with γ = c(Ω)(1 + E(0)^{\frac{q−1}{2q}}).

References

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[2] I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar. 7 (1956), 81-94.

[3] E. H. De Brito, The damped elastic strechted string equation generalized: existence, uniqueness, regularity and stability, Applic. Anal. 13 (1982), 219-233.

[4] R. Ikehata, On the existence of global solutions for some nonlinear hyperbolic equations with Neumann conditions, TRU Math. 24 (1989), 1-17.

[5] R. Ikehata and N. Okazawa, Yosida approximation and nonlinear hyperbolic equations, Nonlinear Analysis T.M.A. 15 (1990), 479-495.

[6] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson-John Wiley, Paris, 1994.