- Open Access
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
- Hyperbolic 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
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). ■
- 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.Google Scholar
- Celebi AO, Kalantarov VK, Tahamtani F: Phragmén-Lindelöf type theorems for some semilinear elliptic and parabolic equations. Demonstratio Mathematica 1998, 31: 43-54.MathSciNetGoogle Scholar
- Flavin JN: On Knowles'version of Saint-Venant's principle in two-dimensional elastostatics. Arch Ration Mech Anal 1974, 53: 366-375. 10.1007/BF00281492View ArticleMathSciNetGoogle Scholar
- Flavin JN, Knops RJ, Payne LE: Decay Estimates for the Constrained Elastic Cylinder of Variable Cross Section. Quart Appl Math 1989, XLVII: 325-350.MathSciNetGoogle Scholar
- 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/BF00946237View ArticleMathSciNetGoogle Scholar
- Flavin JN, Rionero S: Qualitative Estimates for Partial Differential Equations, An Introduction. CRC Press, Roca Raton; 1996.Google Scholar
- 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.MathSciNetGoogle Scholar
- Knowles JK: On Saint-Venant's principle in the two-dimensional linear theory of elasticity. Arch Ration Mech Anal 1966, 21: 123-144.View ArticleGoogle Scholar
- 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.MathSciNetGoogle Scholar
- Horgan CO: Recent developments concerning Saint-Venant's principle: An update. Appl Mech Rev 1989, 42: 295-303. 10.1115/1.3152414View ArticleMathSciNetGoogle Scholar
- Horgan CO: Recent developments concerning Saint-Venant's principle: A second update. Appl Mech Rev 1996, 49: s101-s111. 10.1115/1.3101961View ArticleGoogle Scholar
- 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/BF02415450View ArticleMathSciNetGoogle Scholar
- 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/BF00378164View ArticleMathSciNetGoogle Scholar
- 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)00034-0View ArticleMathSciNetGoogle Scholar
- Quintanilla R: On the spatial blow-up and decay for some nonlinear boundary conditions. Z angew Math Phys 2006, 57: 595-603. 10.1007/s00033-005-0035-4View ArticleMathSciNetGoogle Scholar
- Quintanilla R: Comparison arguments and decay estimates in nonlinear viscoelasticity. Int J Non-linear Mech 2004, 39: 55-61. 10.1016/S0020-7462(02)00127-0View ArticleMathSciNetGoogle Scholar
- 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.2006.06.001View ArticleMathSciNetGoogle Scholar
- Nouria S: Polynomial and analytic boundary feedback stabilization of square plate. Bol Soc Parana Math 2009, 27(2):23-43.Google Scholar
- 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.MathSciNetGoogle Scholar
- Haraux A: Series lacunaires et controle semi-interene des vibrations d'une plaque rectangulaire. J Math Pures App 1989, 68: 457-465.MathSciNetGoogle Scholar
- 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)00080-2View ArticleMathSciNetGoogle Scholar
- 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.MathSciNetGoogle Scholar
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