Abstract
This paper is concerned with investigating the spatial behavior of solutions for a class of hyperbolic equations in semiinfinite cylindrical domains, where nonlinear dissipative boundary conditions imposed on the lateral surface of the cylinder. The main tool used is the weighted energy method.
Mathematics Subject Classification (2010) 35B40, 35L05, 35L35
Keywords:
Hyperbolic equation; Nonlinear boundary conditions; PhragménLindelöf type theorem; Asymptotic behavior1 Introduction
The aim of this paper is to study the spatial asymptotic behavior of solutions of the problem determined by the equation
where a is a positive constant and
where
When we consider equation (1.1), we impose the initial and boundary conditions
where ν is the outward normal to the boundary and
where is a map from R^{+ }into family of bounded domains in R^{n1 }with sufficiently smooth boundary ∂Γ_{τ }such that
In the sequel, we are using
and assume f satisfies
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 [29] and the reviews by Horgan and Knowles [1,10,11]. The energy method is widely used to study such results.
Spatial growth or decay estimates for nontrivial solutions of initial boundary value problems in semiinfinite 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 PhragménLindelöf theorems. In [13], Horgan and Payne proved some these types of theorems and showed the asymptotic behavior of harmonic functions defined on a threedimensional semiinfinite 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 [16,17]).
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 EulerBernoulli plate. For the used methodology, one can see [19,20] where the stabilities are investigated in the cases bounded and unbounded feedbacks for some evolution equations. Recently, Celebi and Kalantarov [21] established a PhragménLindelöf type theorems for a linear wave equation under nonlinear boundary conditions. In our study, we establish PhragménLindelöf 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 ≥ z_{1}, then
(ii) If ψ(z) ≤ cz for some c and z ≥ z_{1}, then
2 Spatial estimates
With the solutions of (1.11.4) with h_{i}(x', t) = 0, i = 1, 2 is naturally associated an energy function
where ._{Ω }denotes the usual norm in L^{2}(Ω).
A multiplication of equation (1.1) by u_{t}, integrating over Ω_{τ }and using (1.31.5):
Since
we obtain
Let δ > 0. Multiplying (1.1) by δu, integrating over Ω_{τ}, and adding to (2.2), we obtain
Integrating (2.3) with respect to t over (0, T) and using (1.5), one can find
On exploiting (2.1) and the inequality , the estimate (2.4) takes the form
by choosing , , . Now we find upper bounds for the right hand side of (2.5). Using the Young's and Schwartz inequalities, we have
By the Poincaré inequality, it is not difficult to see
Inserting (2.9) into (2.8), we get
where Δ' and ∇' are Laplacian and gradient operators in R^{n1}, respectively, Γ_{τ} is the area of Γ_{τ }and λ_{τ }is the Poincaré constant. Now, we recall the inequality
from [13] where and . Using (2.11) and the Hölder's inequality to estimate the boundary integral in (2.10), we obtain
where , , 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
Consequently
where the Young's inequality
for 0 < ε < 1, and γ = μ^{p }have been used. Therefore,
where
By using (2.13) and (2.14), we get
where . From (2.15), it is easy to see
where C is a positive constant.
Next, we exploit Poincaré inequality to estimate
where ρ is the Poincaré constant.
Now, from the inequalities (2.52.7), (2.16), and (2.17), one can find
Upon inserting (2.1) into the right hand side of (2.18), we may write an inequality in the form
At this point, by the inequality (2.19), the function 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
and
where
Theorem 2 Consider the equation (1.1) subject to the conditions u(x', 0, t) = h_{1}(x', t) and for x' ∈ Γ_{0}. If E(+∞) is finite, then
proof By the same manner followed in theorem 1, it is easy to find the inequality
where λ_{τ }is the Poincaré constant. Choosing δ ∈ (0, a), η = min{a δ, δ, 1} and
we obtain
where
Thus, (2.20) follows from (2.21). ■
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the work was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
References

Horgan, CO, Knowles, JK: Recent developments concerning SaintVenant's principle. In: Wu TY, Hutchinson JW (eds.) Advances in Applied Mechanics, vol. 23, pp. 179–269. Academic Press, New York (1983)

Celebi, AO, Kalantarov, VK, Tahamtani, F: PhragménLindelöf type theorems for some semilinear elliptic and parabolic equations. Demonstratio Mathematica. 31, 43–54 (1998)

Flavin, JN: On Knowles'version of SaintVenant's principle in twodimensional elastostatics. Arch Ration Mech Anal. 53, 366–375 (1974). Publisher Full Text

Flavin, JN, Knops, RJ, Payne, LE: Decay Estimates for the Constrained Elastic Cylinder of Variable Cross Section. Quart Appl Math. XLVII, 325–350 (1989)

Flavin, JN, Knops, RJ: Asymptotic behaviour of solutions to semilinear elliptic equations on the half cylinder. Z Angew Math phys. 43, 405–421 (1992). Publisher Full Text

Flavin, JN, Rionero, S: Qualitative Estimates for Partial Differential Equations, An Introduction. CRC Press, Roca Raton (1996)

Horgan, CO: Decay estimates for the biharmonic equation with applications to SaintVenant principles in plane elasticity and Stokes flow. Quart Appl Math. 47, 147–157 (1989)

Knowles, JK: On SaintVenant's principle in the twodimensional linear theory of elasticity. Arch Ration Mech Anal. 21, 123–144 (1966)

Knowles, JK: An energy estimate for the biharmonic equation and its application to SaintVenant's principle in plane elastostatics. Indian J Pure Appl Math. 14, 791–805 (1983)

Horgan, CO: Recent developments concerning SaintVenant's principle: An update. Appl Mech Rev. 42, 295–303 (1989). Publisher Full Text

Horgan, CO: Recent developments concerning SaintVenant's principle: A second update. Appl Mech Rev. 49, s101–s111 (1996). Publisher Full Text

Phragmén, E, Lindelöf, E: Sur une extension d'un principle classique de l'analyse et sur quelque propriétès des functions monogènes dans le voisinage d'un point singulier. Acta Math. 31, 381–406 (1908). Publisher Full Text

Horgan, CO, Payne, LE: PhragménLindelöf Type Results for Harmonic Functions with Nonlinear Boundary Conditions. Arch Rational Mech Anal. 122, 123–144 (1993). Publisher Full Text

Payne, LE, Schaefer, PW, Song, JC: Growth and decay results in heat conduction problems with nonlinear boundary conditions. Nonlinear Anal. 35, 269–286 (1999). Publisher Full Text

Quintanilla, R: On the spatial blowup and decay for some nonlinear boundary conditions. Z angew Math Phys. 57, 595–603 (2006). Publisher Full Text

Quintanilla, R: Comparison arguments and decay estimates in nonlinear viscoelasticity. Int J Nonlinear Mech. 39, 55–61 (2004). Publisher Full Text

Quintanilla, R: PhragménLindelöf alternative for the displacement boundary value problem in a theory of nonlinear micropolar elasticity. Int J Nonlinear Mech. 41, 844–849 (2006). Publisher Full Text

Nouria, S: Polynomial and analytic boundary feedback stabilization of square plate. Bol Soc Parana Math. 27(2), 23–43 (2009)

Ammari, K, Tucsnak, M: Stabilization of second order evolution equations by a class of unbounded feedbacks, ESIM: Control Optim. Calc Var. 6, 361–386 (2001)

Haraux, A: Series lacunaires et controle semiinterene des vibrations d'une plaque rectangulaire. J Math Pures App. 68, 457–465 (1989)

Celebi, AO, Kalantarov, VK: Spatial behaviour estimates for the wave equation under nonlinear boundary condition. Math Comp Model. 34, 527–532 (2001). Publisher Full Text

Ladyzhenskaya, OA, Solonnikov, VA: Determination of solutions of boundary value problems for stationary Stokes and NavierStokes equations having an unbounded Dirichlet integral. Zap Nauch Semin LOMI. 96, 117–160 (1980)