Spatial estimates for a class of hyperbolic equations with nonlinear dissipative boundary conditions
© Tahamtani and Peyravi; licensee Springer. 2011
Received: 3 April 2011
Accepted: 30 August 2011
Published: 30 August 2011
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
KeywordsHyperbolic equation Nonlinear boundary conditions Phragmén-Lindelöf type theorem Asymptotic behavior
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 . 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 , many authors have obtained spatial growth or decay results by Phragmén-Lindelöf theorems. In , 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  proved such results for some classes of heat conduction problems. In , 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  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  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 .
For the proof of our results, we will use the following Lemma.
- (i)If ψ(z) ≤ cz m 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
where ||.||Ω denotes the usual norm in L2(Ω).
where C is a positive constant.
where ρ is the Poincaré constant.
At this point, by the inequality (2.19), the function satisfies in the hypothesis of the Lemma. Therefore, we have proved the following theorem.
Thus, (2.20) follows from (2.21). ■
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