SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Spatial estimates for a class of hyperbolic equations with nonlinear dissipative boundary conditions

Faramarz Tahamtani* and Amir Peyravi

Author affiliations

Department of Mathematics, College of Sciences, Shiraz University, Shiraz, 71454, Iran

For all author emails, please log on.

Citation and License

Boundary Value Problems 2011, 2011:19  doi:10.1186/1687-2770-2011-19


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/19


Received:3 April 2011
Accepted:30 August 2011
Published:30 August 2011

© 2011 Tahamtani and Peyravi; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

Keywords:
Hyperbolic equation; Nonlinear boundary conditions; Phragmén-Lindelöf type theorem; Asymptotic behavior

1 Introduction

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M1">View MathML</a>

(1.1)

where a is a positive constant and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M2">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M3">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M4">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M5">View MathML</a>

(1.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M6">View MathML</a>

(1.4)

where ν is the outward normal to the boundary and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M7">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M8">View MathML</a> is a map from R+ into family of bounded domains in Rn-1 with sufficiently smooth boundary Γτ such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M9">View MathML</a>

In the sequel, we are using

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M10">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M60">View MathML</a>

and assume f satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M11">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M12">View MathML</a>

(1.6)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M13">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M14">View MathML</a>

2 Spatial estimates

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M15">View MathML</a>

(2.1)

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

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M16">View MathML</a>

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M17">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M18">View MathML</a>

(2.2)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M19">View MathML</a>

(2.3)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M20">View MathML</a>

(2.4)

On exploiting (2.1) and the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M21">View MathML</a>, the estimate (2.4) takes the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M22">View MathML</a>

(2.5)

by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M25">View MathML</a>. Now we find upper bounds for the right hand side of (2.5). Using the Young's and Schwartz inequalities, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M26">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M27">View MathML</a>

(2.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M28">View MathML</a>

(2.8)

By the Poincaré inequality, it is not difficult to see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M29">View MathML</a>

(2.9)

Inserting (2.9) into (2.8), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M30">View MathML</a>

(2.10)

where Δ' and ∇' are Laplacian and gradient operators in Rn-1, respectively, |Γτ| is the area of Γτ and λτ is the Poincaré constant. Now, we recall the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M31">View MathML</a>

(2.11)

from [13] where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M33">View MathML</a>. Using (2.11) and the Hölder's inequality to estimate the boundary integral <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M34">View MathML</a> in (2.10), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M35">View MathML</a>

(2.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M37">View MathML</a>, such that r = supτ rτ, λ = infτ λτ, I = supτ Iτ, L = supτ Lτ and m = infτ τ| in which Lτ is the area of ∂Γτ. From (1.6) the inequality (2.12) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M38">View MathML</a>

(2.13)

Consequently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M39">View MathML</a>

where the Young's inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M40">View MathML</a>

for 0 < ε < 1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M41">View MathML</a> and γ = μp have been used. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M42">View MathML</a>

(2.14)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M43">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M44">View MathML</a>

(2.15)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M45">View MathML</a>. From (2.15), it is easy to see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M46">View MathML</a>

where C is a positive constant.

Next, we exploit Poincaré inequality to estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M47">View MathML</a>

(2.17)

where ρ is the Poincaré constant.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M48">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M49">View MathML</a>

(2.19)

At this point, by the inequality (2.19), the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M50">View MathML</a> 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 hi(x', t) = 0, i = 1, 2 under the conditions (1.5) and (1.6). Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M51">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M52">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M53">View MathML</a>

Theorem 2 Consider the equation (1.1) subject to the conditions u(x', 0, t) = h1(x', t) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M54">View MathML</a>for x' ∈ Γ0. If E(+∞) is finite, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M55">View MathML</a>

(2.20)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M56">View MathML</a>

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M57">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M58">View MathML</a>

(2.21)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2011/1/19/mathml/M59">View MathML</a>

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

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

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

  3. Flavin, JN: On Knowles'version of Saint-Venant's principle in two-dimensional elastostatics. Arch Ration Mech Anal. 53, 366–375 (1974). Publisher Full Text OpenURL

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

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

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

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

  8. Knowles, JK: On Saint-Venant's principle in the two-dimensional linear theory of elasticity. Arch Ration Mech Anal. 21, 123–144 (1966)

  9. 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. 14, 791–805 (1983)

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

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

  12. 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 OpenURL

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

  14. 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 OpenURL

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

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

  17. Quintanilla, R: Phragmén-Lindelöf alternative for the displacement boundary value problem in a theory of nonlinear micropolar elasticity. Int J Non-linear Mech. 41, 844–849 (2006). Publisher Full Text OpenURL

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

  19. 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)

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

  21. 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 OpenURL

  22. 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. 96, 117–160 (1980)