- Research
- Open Access
- Published:
Spatial estimates for a class of hyperbolic equations with nonlinear dissipative boundary conditions
Boundary Value Problems volume 2011, Article number: 19 (2011)
Abstract
This paper is concerned with investigating the spatial behavior of solutions for a class of hyperbolic equations in semi-infinite 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
1 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 Rn-1with 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 [2–9] 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 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 [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 Euler-Bernoulli 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 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 limz→∞ψ(z) = ∞. Then φ(z) > 0 satisfying φ(z) < ψ(φ'(z)), z > 0, tends to +∞ when z → +∞.
-
(i)
If ψ(z) ≤ czm for some c and m > 1 for z ≥ z1, then
-
(ii)
If ψ(z) ≤ cz for some c and z ≥ z1, then
2 Spatial estimates
With the solutions of (1.1-1.4) with h i (x', t) = 0, i = 1, 2 is naturally associated an energy function
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):
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 Rn-1, 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 λτ, 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 = μp 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 (2.5-2.7), (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) = h1(x', t) and for x' ∈ Γ0. 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). ■
References
Horgan CO, Knowles JK: Recent developments concerning Saint-Venant's principle. In Advances in Applied Mechanics. Volume 23. Edited by: Wu TY, Hutchinson JW. Academic Press, New York; 1983:179-269.
Celebi AO, Kalantarov VK, Tahamtani F: Phragmén-Lindelöf type theorems for some semilinear elliptic and parabolic equations. Demonstratio Mathematica 1998, 31: 43-54.
Flavin JN: On Knowles'version of Saint-Venant's principle in two-dimensional elastostatics. Arch Ration Mech Anal 1974, 53: 366-375. 10.1007/BF00281492
Flavin JN, Knops RJ, Payne LE: Decay Estimates for the Constrained Elastic Cylinder of Variable Cross Section. Quart Appl Math 1989, XLVII: 325-350.
Flavin JN, Knops RJ: Asymptotic behaviour of solutions to semi-linear elliptic equations on the half cylinder. Z Angew Math phys 1992, 43: 405-421. 10.1007/BF00946237
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 Saint-Venant principles in plane elasticity and Stokes flow. Quart Appl Math 1989, 47: 147-157.
Knowles JK: On Saint-Venant's principle in the two-dimensional linear theory of elasticity. Arch Ration Mech Anal 1966, 21: 123-144.
Knowles JK: An energy estimate for the biharmonic equation and its application to Saint-Venant's principle in plane elastostatics. Indian J Pure Appl Math 1983, 14: 791-805.
Horgan CO: Recent developments concerning Saint-Venant's principle: An update. Appl Mech Rev 1989, 42: 295-303. 10.1115/1.3152414
Horgan CO: Recent developments concerning Saint-Venant's principle: A second update. Appl Mech Rev 1996, 49: s101-s111. 10.1115/1.3101961
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 1908, 31: 381-406. 10.1007/BF02415450
Horgan CO, Payne LE: Phragmén-Lindelöf Type Results for Harmonic Functions with Nonlinear Boundary Conditions. Arch Rational Mech Anal 1993, 122: 123-144. 10.1007/BF00378164
Payne LE, Schaefer PW, Song JC: Growth and decay results in heat conduction problems with nonlinear boundary conditions. Nonlinear Anal 1999, 35: 269-286. 10.1016/S0362-546X(98)00034-0
Quintanilla R: On the spatial blow-up and decay for some nonlinear boundary conditions. Z angew Math Phys 2006, 57: 595-603. 10.1007/s00033-005-0035-4
Quintanilla R: Comparison arguments and decay estimates in nonlinear viscoelasticity. Int J Non-linear Mech 2004, 39: 55-61. 10.1016/S0020-7462(02)00127-0
Quintanilla R: Phragmén-Lindelöf alternative for the displacement boundary value problem in a theory of nonlinear micropolar elasticity. Int J Non-linear Mech 2006, 41: 844-849. 10.1016/j.ijnonlinmec.2006.06.001
Nouria S: Polynomial and analytic boundary feedback stabilization of square plate. Bol Soc Parana Math 2009, 27(2):23-43.
Ammari K, Tucsnak M: Stabilization of second order evolution equations by a class of unbounded feedbacks, ESIM: Control Optim. Calc Var 2001, 6: 361-386.
Haraux A: Series lacunaires et controle semi-interene des vibrations d'une plaque rectangulaire. J Math Pures App 1989, 68: 457-465.
Celebi AO, Kalantarov VK: Spatial behaviour estimates for the wave equation under nonlinear boundary condition. Math Comp Model 2001, 34: 527-532. 10.1016/S0895-7177(01)00080-2
Ladyzhenskaya OA, Solonnikov VA: Determination of solutions of boundary value problems for stationary Stokes and Navier-Stokes equations having an unbounded Dirichlet integral. Zap Nauch Semin LOMI 1980, 96: 117-160.
Author information
Authors and Affiliations
Corresponding author
Additional information
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.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Tahamtani, F., Peyravi, A. Spatial estimates for a class of hyperbolic equations with nonlinear dissipative boundary conditions. Bound Value Probl 2011, 19 (2011). https://doi.org/10.1186/1687-2770-2011-19
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-2770-2011-19
Keywords
- Hyperbolic equation
- Nonlinear boundary conditions
- Phragmén-Lindelöf type theorem
- Asymptotic behavior