非线性粘弹性波动方程解的一致衰减性
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摘要
本文考虑非线性粘弹性波动方程的初边值问题：与非线性粘弹性波动方程的非局部边界耗散问题：解的一致衰减性.在函数g,h和f满足较弱的假设下,通过引入简单的Lyapunov泛函和精确的先验估计证明了,当时间趋于无穷大时上述两个问题的能量泛函以指数形式或多项式形式衰减到零.
In this paper we consider the initial-boundary problem (2.1.1) of a class of nonlinear viscoelastic wave equation and the nonlinear viscoelastic wave equation with nonlocal boundary damping problem Under weaker assumptions on the functions g, h and f, we prove the energy functionals of (2.1.1) and (3.1.1) decay to zero exponentially or polynomially as the time goes to infinity by introducing brief Lyapunov functions and precise priori estimates.
In this paper we consider the initial-boundary problem (2.1.1) of a class of nonlinear viscoelastic wave equation and the nonlinear viscoelastic wave equation with nonlocal boundary damping problem Under weaker assumptions on the functions g, h and f, we prove the energy functionals of (2.1.1) and (3.1.1) decay to zero exponentially or polynomially as the time goes to infinity by introducing brief Lyapunov functions and precise priori estimates.
引文
[1]M. Aassila, M. M. Cavalcant, J. A. Soriano. Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain[J]. SIAM. J. Control Optim., 2000,38(5):1581-1602.
[2]M. Aassila, M.M. Cavalcanti, V. N.Domingos Cavalcanti. Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term[J]. Calc. Var. Partial Differential Equations,2002,15(2):155-180.
[3]C.O. Alves, M.M. Cavalcanti. On existence, u niform decay rates and blow up for solutions of the 2-D wave equation with exponential source[J]. Calc. Var. Partial Differential Equations,2009,34: 377-411.
[4]C. Bardos, G. Lebeau, J. Rauch. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary [J]. SIAM. J. Control Optim.,1992,30:1024-1065.
[5]F. A. Boussouira, P. Cannarsa, D. Sforza. Decay estimates for second order evolution equations with memory [J]. J. Functional Anal.,2008,254:1342-1372.
[6]M.M. Cavalcanti, V.N. Domingos Cavalcanti, P. Martinez. Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term[J]. J. Differential Equations, 2004,203:119-158.
[7]M.M. Cavalcanti, Marcelo M., V.N. Domingos Cavalcanti, I. Lasiecka. Well-posedness and opti-mal decay rates for the wave equation with nonlinear boundary damping-source interaction [J]. J. Differential Equations,2007,236:407-459.
[8]M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A. Soriano. Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result[J]. Trans. Amer. Math. Soc.,2009,361:4561-4580.
[9]M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A.Soriano. Asymptotic stability of the wave equation on compact manifolds and locally distributed damping:a sharp result[J]. Arch. Ration. Mech. Anal.,2010,197:925-964.
[10]M. M. Cavalcanti, V.N. Domingos Cavalcanti, J.S. Prates Filho, J.A. Soriano. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping[J]. Differential Integral Equations,2001,14:85-116.
[11]M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma, J. A. Soriano. Global existence and asymptotic stability for viscoelastic problems[J]. Differential Integral Equations,2002,15:731-748.
[12]M. M. Cavalcanti, H. P. Oquendo. Frictional versus viscoelastic damping in a semilinear wave eqution[J]. SIAM. J. Control Optim.,2003,42(4):1310-1324.
[13]M. M. Cavalcanti, V. N. Domingos Cavalcanti, P. Martinez. General decay rate estimates for viscoelastic dissipative systems[J]. Nonlinear Anal.,2008,68:177-193.
[14]M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.A. Soriano. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping[J]. Electron. J. Differential Equa-tions,2002,2002(44):1-14.
[15]L. C. Evans. Partial differential equations[M]. Grad. Stud. Math., vol19, Amer. Math. Soc., Prov-idence, RI,1998.
[16]V.Georgiev, G. Todorova. Existence of a solution of the wave equation with nonlinear damping and source terms[J]. J. Differential Equations,1994,109:295-308.
[17]I. Lasiecka, D.Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source[J]. Nonlinear Anal.,2008,69:898-910.
[18]Fushan Li, Yuzhen Bai. Uniform rates of decay for nonlinear viscoelastic Marguerre-von Karman shallow shell system[J]. J. Math. Anal. Appl.,2009,351:522-535.
[19]Fushan Li. Limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shells system[J]. J. Differential Equations,2010,249:1241-1257
[20]I. Lasiecka, D. Tataru. Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping[J]. Differential Integral Equations,1993,6:507-533.
[21]S. A. Messaoudi. On the control of solutions of a viscoelastic equation[J]. Journal of the Franklin Institute,2007,344:765-776.
[22]S. A. Messaoudi. General decay of solutions energy in a viscoelastic equation with nonlinear source[J]. Nonlinear Anal.,2008,69:2589-2598.
[23]J.E. Munoz Rivera. Asymptotic behaviour in linear viscoelasticity[J]. Quart. Appl. Math.,1994, 52:628-648.
[24]J.E. Munoz Rivera, J.Barbosa Sobrinho. Existence and uniform rates of decay for contact problems in viscoelasticity[J]. Appl. Anal.,1997,67:175-199.
[25]J.E. Munoz Rivera, A. Peres Salvatierra. Asymptotic behaviour of the energy in partially viscoelas-tic materials[J]. Quart. Appl. Math.,2001,59:557-578.
[26]P. Martinez. A new method to obtain decay rate estimates for dissipative systems with localized damping[J]. Rev. Mat. Complut.,1999,12:251-283.
[27]J.Y. Park, S.H. Park, Busan. Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipative [J]. Czechoslovak Mathematical Journal,2006,56 (131):273-286.
[28]M. Nakao. A difference inequality and its applications to nonlinear evolution equations[J]. J. Math. Soc. Japan,1978,30:747-762.
[29]M. Nakao. Decay of solutions of the wave equation with a local nonlinear dissipation[J]. Math. Ann.,1996,305:403-417.
[30]M. A. Rammaha. The influence of damping and source terms on solutions of nonlinear wave equations[J]. Bol. Soc. Parana. Mat.,2007,25(3):77-90.
[31]M. L. Santos. Decay rates for solutions of a system of wave equations with memory[J]. Electron. J. Differential Equations,2002, (44):1-17.
[32]G. Todorova. Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms [J]. Nonlinear Anal.,2000,41:891-905.
[33]D. Toundykov. Optimal decay rates for solutions of a nonlinear wave equation with localized non-linear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions [J]. Nonlinear Anal.,2007,67:512-544.
[34]E. Zuazua. Stability and decay for a class of nonlinear hyperbolic problems[J]. Asymptot. Anal., 1988,1:161-185.
[35]E. Zuazua. Exponential decay for the semilinear wave equation with locally distributed damping[J]. Comm. Partial Differential Equations,1990,15:205-235.
[2]M. Aassila, M.M. Cavalcanti, V. N.Domingos Cavalcanti. Existence and uniform decay of the wave equation with nonlinear boundary damping and boundary memory source term[J]. Calc. Var. Partial Differential Equations,2002,15(2):155-180.
[3]C.O. Alves, M.M. Cavalcanti. On existence, u niform decay rates and blow up for solutions of the 2-D wave equation with exponential source[J]. Calc. Var. Partial Differential Equations,2009,34: 377-411.
[4]C. Bardos, G. Lebeau, J. Rauch. Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary [J]. SIAM. J. Control Optim.,1992,30:1024-1065.
[5]F. A. Boussouira, P. Cannarsa, D. Sforza. Decay estimates for second order evolution equations with memory [J]. J. Functional Anal.,2008,254:1342-1372.
[6]M.M. Cavalcanti, V.N. Domingos Cavalcanti, P. Martinez. Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term[J]. J. Differential Equations, 2004,203:119-158.
[7]M.M. Cavalcanti, Marcelo M., V.N. Domingos Cavalcanti, I. Lasiecka. Well-posedness and opti-mal decay rates for the wave equation with nonlinear boundary damping-source interaction [J]. J. Differential Equations,2007,236:407-459.
[8]M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A. Soriano. Asymptotic stability of the wave equation on compact surfaces and locally distributed damping-a sharp result[J]. Trans. Amer. Math. Soc.,2009,361:4561-4580.
[9]M. M. Cavalcanti, V. N. Domingos Cavalcanti, R. Fukuoka, J. A.Soriano. Asymptotic stability of the wave equation on compact manifolds and locally distributed damping:a sharp result[J]. Arch. Ration. Mech. Anal.,2010,197:925-964.
[10]M. M. Cavalcanti, V.N. Domingos Cavalcanti, J.S. Prates Filho, J.A. Soriano. Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping[J]. Differential Integral Equations,2001,14:85-116.
[11]M. M. Cavalcanti, V. N. Domingos Cavalcanti, T. F. Ma, J. A. Soriano. Global existence and asymptotic stability for viscoelastic problems[J]. Differential Integral Equations,2002,15:731-748.
[12]M. M. Cavalcanti, H. P. Oquendo. Frictional versus viscoelastic damping in a semilinear wave eqution[J]. SIAM. J. Control Optim.,2003,42(4):1310-1324.
[13]M. M. Cavalcanti, V. N. Domingos Cavalcanti, P. Martinez. General decay rate estimates for viscoelastic dissipative systems[J]. Nonlinear Anal.,2008,68:177-193.
[14]M.M. Cavalcanti, V.N. Domingos Cavalcanti, J.A. Soriano. Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping[J]. Electron. J. Differential Equa-tions,2002,2002(44):1-14.
[15]L. C. Evans. Partial differential equations[M]. Grad. Stud. Math., vol19, Amer. Math. Soc., Prov-idence, RI,1998.
[16]V.Georgiev, G. Todorova. Existence of a solution of the wave equation with nonlinear damping and source terms[J]. J. Differential Equations,1994,109:295-308.
[17]I. Lasiecka, D.Toundykov, Regularity of higher energies of wave equation with nonlinear localized damping and a nonlinear source[J]. Nonlinear Anal.,2008,69:898-910.
[18]Fushan Li, Yuzhen Bai. Uniform rates of decay for nonlinear viscoelastic Marguerre-von Karman shallow shell system[J]. J. Math. Anal. Appl.,2009,351:522-535.
[19]Fushan Li. Limit behavior of the solution to nonlinear viscoelastic Marguerre-von Karman shallow shells system[J]. J. Differential Equations,2010,249:1241-1257
[20]I. Lasiecka, D. Tataru. Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping[J]. Differential Integral Equations,1993,6:507-533.
[21]S. A. Messaoudi. On the control of solutions of a viscoelastic equation[J]. Journal of the Franklin Institute,2007,344:765-776.
[22]S. A. Messaoudi. General decay of solutions energy in a viscoelastic equation with nonlinear source[J]. Nonlinear Anal.,2008,69:2589-2598.
[23]J.E. Munoz Rivera. Asymptotic behaviour in linear viscoelasticity[J]. Quart. Appl. Math.,1994, 52:628-648.
[24]J.E. Munoz Rivera, J.Barbosa Sobrinho. Existence and uniform rates of decay for contact problems in viscoelasticity[J]. Appl. Anal.,1997,67:175-199.
[25]J.E. Munoz Rivera, A. Peres Salvatierra. Asymptotic behaviour of the energy in partially viscoelas-tic materials[J]. Quart. Appl. Math.,2001,59:557-578.
[26]P. Martinez. A new method to obtain decay rate estimates for dissipative systems with localized damping[J]. Rev. Mat. Complut.,1999,12:251-283.
[27]J.Y. Park, S.H. Park, Busan. Existence and asymptotic stability for viscoelastic problems with nonlocal boundary dissipative [J]. Czechoslovak Mathematical Journal,2006,56 (131):273-286.
[28]M. Nakao. A difference inequality and its applications to nonlinear evolution equations[J]. J. Math. Soc. Japan,1978,30:747-762.
[29]M. Nakao. Decay of solutions of the wave equation with a local nonlinear dissipation[J]. Math. Ann.,1996,305:403-417.
[30]M. A. Rammaha. The influence of damping and source terms on solutions of nonlinear wave equations[J]. Bol. Soc. Parana. Mat.,2007,25(3):77-90.
[31]M. L. Santos. Decay rates for solutions of a system of wave equations with memory[J]. Electron. J. Differential Equations,2002, (44):1-17.
[32]G. Todorova. Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms [J]. Nonlinear Anal.,2000,41:891-905.
[33]D. Toundykov. Optimal decay rates for solutions of a nonlinear wave equation with localized non-linear dissipation of unrestricted growth and critical exponent source terms under mixed boundary conditions [J]. Nonlinear Anal.,2007,67:512-544.
[34]E. Zuazua. Stability and decay for a class of nonlinear hyperbolic problems[J]. Asymptot. Anal., 1988,1:161-185.
[35]E. Zuazua. Exponential decay for the semilinear wave equation with locally distributed damping[J]. Comm. Partial Differential Equations,1990,15:205-235.