The aim of this paper is to study the spatial asymptotic behavior of solutions of the problem determined by the equation

u t t = Δ u t - a u t - Δ 2 u , ( x , t ) ∈ Ω × ( 0 , ∞ ) ,

where a is a positive constant and

Ω = { x ∈ R n : x n ∈ R + , x ′ = ( x 1 , … , x n - 1 ) ∈ Γ x n ⊂ R n - 1 } ,

where

Γ τ = { ( x ′ , x n ) ∈ Ω : x n = τ } .

When we consider equation (1.1), we impose the initial and boundary conditions

u ( x , 0 ) = u t ( x , 0 ) = 0 , x ∈ Ω ,

u ( x ′ , 0 , t ) = h 1 ( x ′ , t ) , ∂ u ∂ v ( x ′ , 0 , t ) = h 2 ( x ′ , t ) , ( x ′ , t ) ∈ Γ 0 × ( 0 , ∞ ) ,

u = 0 , Δ u = - f ∂ u ∂ ν , ( x , t ) ∈ Σ 0 × ( 0 , ∞ ) ,

where ν is the outward normal to the boundary and

Σ τ = { x ∈ R n : x ′ ∈ ∂ Γ x n , τ ≤ x n < ∞ } ,

where τ → Γ ̄ τ is a map from R⁺ into family of bounded domains in R^n-1 with sufficiently smooth boundary ∂Γ_τ such that

0 < m 0 ≤ inf τ Γ τ ≤ sup τ Γ τ ≤ m 1 < ∞ .

In the sequel, we are using

Ω τ = Ω ∩ { x ∈ R n : 0 < x n < τ } ,

R τ = Ω ∩ { x ∈ R n : τ < x n < ∞ } ,

and assume f satisfies

F ( v ) = ∫ 0 v f ( ξ ) d ξ ≥ α v f ( v ) > 0 , α > 0 , ∀ v ∈ R ,

v f ( v ) ≥ γ v 2 p , p > 1 2 , γ > 0 , ∀ v ∈ R .

In recent years, much attention has been directed to the study of spatial behavior of solutions of partial differential equations and systems. The history and development of this question is explained in the work of Horgan and Knowles 1. The interested reader is referred to the papers 23456789 and the reviews by Horgan and Knowles 11011. The energy method is widely used to study such results.

Spatial growth or decay estimates for nontrivial solutions of initial -boundary value problems in semi-infinite domains with nonlinearities on the boundary have been studied by many authors. Since 1908, when Edvard Phragmén and Ernst Lindelöf published their idea 12, many authors have obtained spatial growth or decay results by Phragmén-Lindelöf theorems. In 13, Horgan and Payne proved some these types of theorems and showed the asymptotic behavior of harmonic functions defined on a three-dimensional semi-infinite cylinder when homogeneous nonlinear boundary conditions are imposed on the lateral surface of the cylinder. Payne and Schaefer 14 proved such results for some classes of heat conduction problems. In 15, Quintanilla investigate the spatial behavior of several nonlinear parabolic equations with nonlinear boundary conditions, (see also 1617).

Under nonlinear dissipative feedbacks on the boundary, Nouria 18 proved a polynomial stability for regular initial data and exponential stability for some analytic initial data of a square Euler-Bernoulli plate. For the used methodology, one can see 1920 where the stabilities are investigated in the cases bounded and unbounded feedbacks for some evolution equations. Recently, Celebi and Kalantarov 21 established a Phragmén-Lindelöf type theorems for a linear wave equation under nonlinear boundary conditions. In our study, we establish Phragmén-Lindelöf type theorems for equation (1.1) with nonlinear dissipative feedback terms on the boundary. Our study is inspired by the results of 21.

For the proof of our results, we will use the following Lemma.

Lemma 22 Let ψ be a monotone increasing function with ψ(0) = 0 and lim_z→∞ψ(z) = ∞. Then φ(z) > 0 satisfying φ(z) < ψ(φ'(z)), z > 0, tends to +∞ when z → +∞.

(i) If ψ(z) ≤ cz^m for some c and m > 1 for z ≥ z₁, then

liminf z → + ∞ z - m m - 1 φ ( z ) > 0 .

(ii) If ψ(z) ≤ cz for some c and z ≥ z₁, then

liminf z → + ∞ φ ( z ) exp - z c > 0 .

With the solutions of (1.1-1.4) with h_i(x', t) = 0, i = 1, 2 is naturally associated an energy function

E ( τ ) = ∫ 0 T | | u t | | Ω τ 2 + | | ∇ u t | | Ω τ 2 + | | Δ u | | Ω τ 2 + ∫ 0 τ ∫ ∂ Γ η ∇ u f ( ∇ u ) d s d η d t ,

where ||.||_Ω denotes the usual norm in L²(Ω).

A multiplication of equation (1.1) by u_t, integrating over Ω_τ and using (1.3-1.5):

d d t 1 2 | | u t | | Ω τ 2 + 1 2 | | Δ u | | Ω τ 2 + ∫ 0 τ ∫ ∂ Γ η F ( ∇ u ) d s d η + a | | u t | | Ω τ 2 + | | ∇ u t | | Ω τ 2 = - ( u t , u x n x n x n ) Γ τ + ( u t x n , u x n x n ) Γ τ + ( u t , u t x n ) Γ τ .

Since

( u t , u x n x n x n ) Γ τ = - ( u t x n , u x n x n ) Γ τ ,

we obtain

d d t 1 2 | | u t | | Ω τ 2 + 1 2 | | Δ u | | Ω τ 2 + ∫ 0 τ ∫ ∂ Γ η F ( ∇ u ) d s d η + a | | u t | | Ω τ 2 + | | ∇ u t | | Ω τ 2 = 2 ( u t x n , u x n x n ) Γ τ + ( u t , u t x n ) Γ τ .

Let δ > 0. Multiplying (1.1) by δu, integrating over Ω_τ, and adding to (2.2), we obtain

d d t 1 2 | | u t | | Ω τ 2 + 1 2 | | Δ u | | Ω τ 2 + δ ( u , u t ) Ω τ + a δ 2 | | u | | Ω τ 2 + δ 2 | | ∇ u | | Ω τ 2 + ∫ 0 τ ∫ ∂ Γ η F ( ∇ u ) d s d η + ( a - δ ) | | u t | | Ω τ 2 + | | ∇ u t | | Ω τ 2 + δ | | Δ u | | Ω τ 2 + δ ∫ 0 τ ∫ ∂ Γ η ∇ u f ( ∇ u ) d s d η = 2 ( u t x n , u x n x n ) Γ τ + ( u t , u t x n ) Γ τ + 2 δ ( u x n , u x n x n ) Γ τ + δ ( u , u t x n ) Γ τ .

Integrating (2.3) with respect to t over (0, T) and using (1.5), one can find

1 2 | | u t | | Ω τ 2 +   1 2 | | Δ u | | Ω τ 2 +   δ 2 | | ∇ u | | Ω τ 2 +   a δ 2 | | u | | Ω τ 2       + δ ( u , u t ) Ω τ + α ∫ 0 τ ∫ ∂ Γ η ∇ u f ( ∇ u ) d s d η       + ( a − δ ) ∫ 0 T | | u t | | Ω τ 2 d t + δ ∫ 0 T | | Δ u | | Ω τ 2 d t + ∫ 0 T | | ∇ u t | | Ω τ 2 d t       + δ ∫ 0 T ∫ 0 τ ∫ ∂ Γ η ∇ u f ( ∇ u ) d s d η d t ≤ ∫ 0 T [ 2 ( u tx n , u x n x n ) Γ τ + ( u t , u tx n ) Γ τ ]   dt         + ∫ 0 T [ 2 δ ( u x n , u x n x n ) Γ τ + δ ( u , u t x n ) Γ τ ]   dt .

On exploiting (2.1) and the inequality - 1 4 u t Ω τ 2 - δ 2 u Ω τ 2 ≤ δ ( u , u t ) Ω τ , the estimate (2.4) takes the form

σ - 1 E ( τ ) ≤ ∫ 0 T [ ( u t x n , u x n x n ) Γ τ + ( u t , u t x n ) Γ τ ] d t + ∫ 0 T [ ( u x n , u x n x n ) Γ τ + ( u , u t x n ) Γ τ ] d t ,

by choosing δ = a 2 , δ 1 = min { 1 , a 2 } , σ = max { a δ 1 , 2 δ 1 } . Now we find upper bounds for the right hand side of (2.5). Using the Young's and Schwartz inequalities, we have

∫ 0 T ( u t x n , u x n x n ) Γ τ d t ≤ 1 2 ∫ 0 T | | ∇ u t | | Γ τ 2 d t + 1 2 ∫ 0 T | | Δ u | | Γ τ 2 d t ,

∫ 0 T ( u t , u t x n ) Γ τ d t ≤ 1 2 ∫ 0 T | | u t | | Γ τ 2 d t + 1 2 ∫ 0 T | | ∇ u t | | Γ τ 2 d t ,

∫ 0 T ( u x n , u x n x n ) Γ τ d t ≤ ∫ 0 T | | u x n | | Γ τ | | u x n x n | | Γ τ d t .

By the Poincaré inequality, it is not difficult to see

v D 2 ≤ λ - 1 ∇ v D 2 + D - 1 ∫ D v d A 2 .

Inserting (2.9) into (2.8), we get

∫ 0 T ( u x n , u x n x n ) Γ τ d t ≤ ∫ 0 T λ τ - 1 2 | | Δ ′ u | | Γ τ + Γ τ - 1 2 ∫ Γ τ ∇ ′ u d A | | u x n x n | | Γ τ d t ,

where Δ' and ∇' are Laplacian and gradient operators in R^n-1, respectively, |Γ_τ| is the area of Γ_τ and λ_τ is the Poincaré constant. Now, we recall the inequality

∫ D v d A ≤ r 0 2 ∫ ∂ D v d s + I 0 1 2 2 ∫ D ∇ v 2 d A 1 2 ,

from 13 where r 0 2 = sup D | x ′ | 2 and I 0 = ∫ D | x ′ | 2 d A . Using (2.11) and the Hölder's inequality to estimate the boundary integral ∫ Γ τ ∇ ′ u d A in (2.10), we obtain

∫ 0 T ( u x n , u x n x n ) Γ τ d t ≤ ∫ 0 T M 1 | | Δ ′ u | | Γ τ + γ 1 2 p M 2 ∫ ∂ Γ τ | ∇ ′ u | 2 p d A 1 2 p | | u x n x n | | Γ τ d t ,

where M 1 = λ - 1 2 + I 1 ∕ 2 2 m 1 ∕ 2 , M 2 = 1 2 r L ( 2 p - 1 ) ∕ 2 p m - 1 ∕ 2 γ - 1 ∕ 2 p , such that r = sup_τ r_τ, λ = inf_τ λ_τ, I = sup_τ I_τ, L = sup_τ L_τ and m = inf_τ |Γ_τ| in which L_τ is the area of ∂Γ_τ. From (1.6) the inequality (2.12) yields

∫ 0 T ( u x n , u x n x n ) Γ τ d t ≤ M 1 ∫ 0 T | | Δ u | | Γ τ 2 d t + M 2 ∫ 0 T ∫ ∂ Γ τ ∇ ′ u f ( ∇ ′ u ) d A 1 2 p | | u x n x n | | Γ τ d t .

Consequently

∫ ∂ Γ τ ∇ ′ u f ( ∇ ′ u ) d s 1 2 p ∫ Γ τ u x n x n 2 d A 1 2 = ∫ ∂ Γ τ ∇ ′ u f ( ∇ ′ u ) d s 1 p + 1 ∫ Γ τ u x n x n 2 d A p p + 1 p + 1 2 p ≤ μ p 1 + p ∫ ∂ Γ τ ∇ ′ u f ( ∇ ′ u ) d s + p μ ( 1 + p ) ∫ Γ τ u x n x n 2 d A p + 1 2 p ,

where the Young's inequality

α ε β 1 - ε = ( α γ ) ε β γ - ε 1 - ε ( 1 - ε ) ≤ ε α γ + ( 1 - ε ) β γ - ε 1 - ε ,

for 0 < ε < 1, μ = p 1 p + 1 and γ = μ^p have been used. Therefore,

∫ ∂ Γ τ ∇ ′ u f ( ∇ ′ u ) d s 1 2 p ∫ Γ τ u x n x n 2 d A 1 2 ≤ N ( p ) ∫ ∂ Γ τ ∇ ′ u f ( ∇ ′ u ) d s + ∫ Γ τ u x n x n 2 d A p + 1 2 p ,

where

N ( p ) = p p p + 1 ( 1 + p ) .

By using (2.13) and (2.14), we get

∫ 0 T ( u x n , u x n x n ) Γ τ d t ≤ M 1 ∫ 0 T | | Δ u | | Γ τ 2 d t       + M 2 N ˜ ( p ) ∫ 0 T ( ∫ ∂ Γ τ ∇ ′ u f ( ∇ ′ u ) d s + ∫ Γ τ u x n x n 2 d A ) p + 1 2 p d t ,

where N ˜ ( p ) = [ N ( p ) ] p + 1 2 p . From (2.15), it is easy to see

∫ 0 T ( u x n , u x n x n ) Γ τ d t ≤ M 1 ∫ 0 T | | Δ u | | Γ τ 2 d t       + M 2 C N ˜ ( p ) ( ∫ 0 T ∫ ∂ Γ τ ∇ u f ( ∇ u ) d s d t + ∫ 0 T | | Δ u | | Γ τ 2 d t ) p + 1 2 p ,

where C is a positive constant.

Next, we exploit Poincaré inequality to estimate

∫ 0 T ( u , u t x n ) Γ τ d t ≤ ρ - 1 2 ∫ 0 T | | Δ u | | Γ τ 2 d t + 1 2 ∫ 0 T | | ∇ u t | | Γ τ 2 d t ,

where ρ is the Poincaré constant.

Now, from the inequalities (2.5-2.7), (2.16), and (2.17), one can find

E ( τ ) ≤ ∫ 0 T [ σ 2 | | u t | | Γ τ 2 + 3 2 σ | | ∇ u t | | Γ τ 2 + σ ( 1 2 + M 1 + ρ − 1 2 ) | | Δ u | | Γ τ 2 ] d t       + ∫ 0 T ∫ ∂ Γ τ ∇ u f ( ∇ u ) d s d t + σ M 2 C N ˜ ( p ) { ∫ 0 T ∫ ∂ Γ τ ∇ u f ( ∇ u ) d s d t       + ∫ 0 T [ | | ∇ u t | | Γ τ 2 d t + | | Δ u | | Γ τ 2 + | | u t | | Γ τ 2 ] d t } p + 1 2 p .

Upon inserting (2.1) into the right hand side of (2.18), we may write an inequality in the form

E ( τ ) ≤ σ ( 5 2 + M 1 + ρ − 1 2 ) E ′ ( τ ) + σ M 2 C N ˜ ( p )   [ E ′ ( τ ) ] p + 1 2 p .

At this point, by the inequality (2.19), the function ψ ( z ) = α 1 z + α 2 z p + 1 2 p satisfies in the hypothesis of the Lemma. Therefore, we have proved the following theorem.

Theorem 1 Let u(x, t) be a nontrivial solution of (1.1) - (1.4) with h_i(x', t) = 0, i = 1, 2 under the conditions (1.5) and (1.6). Then

lim inf τ → + ∞ E ( τ ) τ − p + 1 1 − p > 0 ,       p ∈ ( 1 2 , 1 ) ,

and

lim inf τ → + ∞ E ( τ ) exp ( − τ c ) > 0 ,       p ∈ [ 1 , + ∞ ) ,

where

c = max { σ ( 5 2 + M 1 + ρ − 1 2 ) , σ M 2 C N ˜ ( p ) } .

Theorem 2 Consider the equation (1.1) subject to the conditions u(x', 0, t) = h₁(x', t) and ∂ u ∂ v ( x ′ , 0 , t ) = h 2 ( x ′ , t ) for x' ∈ Γ₀. If E(+∞) is finite, then

lim τ → + ∞ ∫ 0 T | | u t | | R τ 2 d t + ∫ 0 T | | ∇ u t | | R τ 2 d t + ∫ 0 T | | Δ u | | R τ 2 d t = 0 .

proof By the same manner followed in theorem 1, it is easy to find the inequality

( a - δ ) ∫ 0 T | | u t | | R τ 2 d t + ∫ 0 T | | ∇ u t | | R τ 2 d t + δ ∫ 0 T | | Δ u | | R τ 2 d t ≤ 1 2 ∫ 0 T | | u t | | Γ τ 2 d t + 3 2 + δ 2 ∫ 0 T | | ∇ u t | | Γ τ 2 d t + [ 1 + δ ( 1 + λ τ - 1 + 1 2 λ τ - 2 ) ] ∫ 0 T | | Δ u | | Γ τ 2 d t ,

where λ_τ is the Poincaré constant. Choosing δ ∈ (0, a), η = min{a -δ, δ, 1} and

γ ̃ = η - 1 max { 3 2 + δ 2 , 1 + δ ( 1 + λ τ - 1 + 1 2 λ τ - 2 ) } ,

we obtain

Ẽ ( τ ) ≤ - γ ̃ Ẽ ′ ( τ ) ,

where

Ẽ ( τ ) = ∫ 0 T | | u t | | R τ 2 d t + ∫ 0 T | | ∇ u t | | R τ 2 d t + ∫ 0 T | | Δ u | | R τ 2 d t .

Thus, (2.20) follows from (2.21).   ■

The authors declare that they have no competing interests.

The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.