On Coupled Klein-Gordon-Schrödinger Equations with Acoustic Boundary Conditions
© Tae Gab Ha and Jong Yeoul Park. 2010
Received: 8 July 2010
Accepted: 10 September 2010
Published: 15 September 2010
We are concerned with the existence and energy decay of solution to the initial boundary value problem for the coupled Klein-Gordon-Schrödinger equations with acoustic boundary conditions.
where is a bounded domain, , with boundary of class , where and are two disjoint pieces of each having nonempty interior and are given functions. We will denote by the unit outward normal vector to . stands for the Laplacian with respect to the spatial variables; denotes the derivative with respect to time . Here is the normal displacement to the boundary at time with the boundary point .
The above equations describe a generalization of the classical model of the Yukawa interaction of conserved complex nucleon field with neutral real meson field. Here, is complex scalar nucleon field while and are real scalar meson one.
Boundary conditions of the fifth and sixth equations are called acoustic boundary conditions. Equation (1.1)5 (the fifth equation of (1.1)) does not contain the second derivative , which physically means that the material of the surface is much more lighter than a liquid flowing along it. As far as in (1.1)6 (the sixth equation of (1.1)) is concerned to the case of a nonporous boundary, (1.1)6 simulates a porous boundary when a function is nonnegative. When general acoustic boundary conditions, which had the presence of in (1.1)5, are prescribed on the whole boundary, Beale [12–14] proved the global existence and regularity of solutions in a Hilbert space of data with finite energy by means of semigroup methods. The asymptotic behavior was obtained in , but no decay rate was given there. Recently, the acoustic boundary conditions have been treated by many authors (cf. [15–21] and a list of references therein). However, energy decay problem with acoustic boundary conditions was studied by a few authors. For instance, Rivera and Qin  proved the polynomial decay for the energy of the wave motion using the Lyapunov functional technique in the case of general acoustic boundary conditions and . Frota and Larkin  considered global solvability and the exponential decay of the energy for the wave equation with acoustic boundary conditions, which eliminated the second derivative term for . However, it is not simple to apply the semigroup theory as well as Galerkin's method in  because a system of corresponding ordinary equations is not normal and one cannot apply directly the Carathéodory's theorem. So they overcame this problem using the degenerated second order equation. And Park and Ha  studied the existence and uniqueness of solutions and uniform decay rates for the Klein-Gordon-type equation by using the multiplier technique. Moreover,  proved the exponential and polynomial decay rates of solutions for all .
In this paper, we prove the existence and uniqueness of solutions and uniform decay rates for the Klein-Gordon-Schrödinger equations with acoustic boundary conditions and allow to apply the method developed in . However,  did not treat the Klein-Gordon-Schrödinger equations.
This paper is organized as follows. In Section 2, we recall the notation and hypotheses and introduce our main result. In Section 3, using Galerkin's method, we prove the existence and uniqueness of solutions to problem (1.1). In Section 4, we prove the exponential energy decay rate for the solutions obtained in Section 3.
2. Notations and Main Results
In physical situation, and are parameters representing the gratitude of diffusion and dissipation effects. Also, is a fluid density and describes the mass of a meson. Boundary condition (1.1)6 (the sixth equation of (1.1)) simulates a porous boundary because of the function .
Now, we are in a position to state our main result.
3. Existence of Solutions
where and for all , , . The local existence of regular functions , , and is standard, because (3.2) is a normal system of ordinary differential equation. A solution to the problem (1.1) on some interval will be obtained as the limit of as and . Then, this solution can be extended to the whole interval , for all , as a consequence of the a priori estimates that will be proved in the next step.
3.1. The First Estimate
3.2. The Second Estimate
3.3. The Third Estimate
Thus, by the above convergences and (3.53), we can prove the existence of solutions to (1.1) satisfying (2.8).
4. Uniform Decay
Now, we will estimate the terms on the right-hand side of (4.14).
We now estimate the last term on the right-hand side of (4.20).
This implies the proof of Theorem 2.1 is completed.
- Bachelot A: Problème de Cauchy pour des systèmes hyperboliques semilinéaires. Annales de l'Institut Henri Poincaré. Analyse Non Linéaire 1984,1(6):453-478.MathSciNetMATHGoogle Scholar
- Baillon J-B, Chadam JM: The Cauchy problem for the coupled Schrödinger-Klein-Gordon equations. In Contemporary Developments in Continuum Mechanics and Partial Differential Equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Jane, North-Holland Mathematics Studies. Volume 30. Edited by: Medeiros. North-Holland, Amsterdam, The Netherlands; 1978:37-44.Google Scholar
- Fukuda I, Tsutsumi M: On coupled Klein-Gordon-Schrödinger equations. I. Bulletin of Science and Engineering Research Laboratory. Waseda University 1975, (69):51-62.Google Scholar
- Fukuda I, Tsutsumi M: On coupled Klein-Gordon-Schrödinger equations. II. Journal of Mathematical Analysis and Applications 1978,66(2):358-378. 10.1016/0022-247X(78)90239-1MathSciNetView ArticleMATHGoogle Scholar
- Fukuda I, Tsutsumi M: On the Yukawa-coupled Klein-Gordon-Schrödinger equations in three space dimensions. Proceedings of the Japan Academy 1975,51(6):402-405. 10.3792/pja/1195518563MathSciNetView ArticleMATHGoogle Scholar
- Bisognin V, Cavalcanti MM, Cavalcanti VND, Soriano J: Uniform decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Nonlinear Differential Equations and Applications 2008,15(1-2):91-113. 10.1007/s00030-007-6025-9MathSciNetView ArticleMATHGoogle Scholar
- Cavalcanti MM, Cavalcanti VND: Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations. Nonlinear Differential Equations and Applications 2000,7(3):285-307. 10.1007/PL00001426MathSciNetView ArticleMATHGoogle Scholar
- Colliander J, Holmer J, Tzirakis N: Low regularity global well-posedness for the Zakharov and Klein-Gordon-Schrödinger systems. Transactions of the American Mathematical Society 2008,360(9):4619-4638. 10.1090/S0002-9947-08-04295-5MathSciNetView ArticleMATHGoogle Scholar
- Miao C, Xu G:Global solutions of the Klein-Gordon-Schrödinger system with rough data in . Journal of Differential Equations 2006,227(2):365-405. 10.1016/j.jde.2005.10.012MathSciNetView ArticleMATHGoogle Scholar
- Park JY, Kim JA: Global existence and uniform decay for the coupled Klein-Gordon-Schrödinger equations with non-linear boundary damping and memory term. Mathematical Methods in the Applied Sciences 2006,29(9):947-964. 10.1002/mma.673MathSciNetView ArticleMATHGoogle Scholar
- Tzirakis N: The Cauchy problem for the Klein-Gordon-Schrödinger system in low dimensions below the energy space. Communications in Partial Differential Equations 2005,30(4–6):605-641.MathSciNetView ArticleMATHGoogle Scholar
- Beale JT: Spectral properties of an acoustic boundary condition. Indiana University Mathematics Journal 1976,25(9):895-917. 10.1512/iumj.1976.25.25071MathSciNetView ArticleMATHGoogle Scholar
- Beale JT: Acoustic scattering from locally reacting surfaces. Indiana University Mathematics Journal 1977,26(2):199-222. 10.1512/iumj.1977.26.26015MathSciNetView ArticleMATHGoogle Scholar
- Beale JT, Rosencrans SI: Acoustic boundary conditions. Bulletin of the American Mathematical Society 1974, 80: 1276-1278. 10.1090/S0002-9904-1974-13714-6MathSciNetView ArticleMATHGoogle Scholar
- Cousin AT, Frota CL, Larkin NA: Global solvability and asymptotic behaviour of a hyperbolic problem with acoustic boundary conditions. Funkcialaj Ekvacioj 2001,44(3):471-485.MathSciNetMATHGoogle Scholar
- Cousin AT, Frota CL, Larkin NA: On a system of Klein-Gordon type equations with acoustic boundary conditions. Journal of Mathematical Analysis and Applications 2004,293(1):293-309. 10.1016/j.jmaa.2004.01.007MathSciNetView ArticleMATHGoogle Scholar
- Frota CL, Goldstein JA: Some nonlinear wave equations with acoustic boundary conditions. Journal of Differential Equations 2000,164(1):92-109. 10.1006/jdeq.1999.3743MathSciNetView ArticleMATHGoogle Scholar
- Ha TG, Park JY: Existence of solutions for the Kirchhoff-type wave equation with memory term and acoustic boundary conditions. Numerical Functional Analysis and Optimization 2010,31(8):921-935. 10.1080/01630563.2010.498301MathSciNetView ArticleMATHGoogle Scholar
- Mugnolo D: Abstract wave equations with acoustic boundary conditions. Mathematische Nachrichten 2006,279(3):299-318. 10.1002/mana.200310362MathSciNetView ArticleMATHGoogle Scholar
- Park JY, Ha TG: Well-posedness and uniform decay rates for the Klein-Gordon equation with damping term and acoustic boundary conditions. Journal of Mathematical Physics 2009,50(1):-18.MathSciNetView ArticleMATHGoogle Scholar
- Vicente A: Wave equation with acoustic/memory boundary conditions. Boletim da Sociedade Paranaense de Matemática 2009,27(1):29-39.MathSciNetView ArticleMATHGoogle Scholar
- Muñoz Rivera JE, Qin Y: Polynomial decay for the energy with an acoustic boundary condition. Applied Mathematics Letters 2003,16(2):249-256. 10.1016/S0893-9659(03)80039-3MathSciNetView ArticleMATHGoogle Scholar
- Frota CL, Larkin NA: Uniform stabilization for a hyperbolic equation with acoustic boundary conditions in simple connected domains. In Contributions to Nonlinear Analysis, Progress in Nonlinear Differential Equations and Their Applications. Volume 66. Birkhäuser, Basel, Switzerland; 2006:297-312. 10.1007/3-7643-7401-2_20Google Scholar
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