September  2021, 14(9): 3141-3166. doi: 10.3934/dcdss.2021077

Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels

1. 

School of Science, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

2. 

School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China

3. 

Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

* Corresponding author: Ti-Jun Xiao

Received  September 2020 Revised  January 2021 Published  September 2021 Early access  June 2021

Fund Project: The second author is supported by NSF of China Grant No. 11971306. The third author is supported by NSF of China Grant No. 11771091 and No. 11831011

We are concerned with the polynomial stability and the integrability of the energy for second order integro-differential equations in Hilbert spaces with positive definite kernels, where the memory can be oscillating or sign-varying or not locally absolutely continuous (without any control conditions on the derivative of the kernel). For this stability problem, tools from the theory of existing positive definite kernels can not be applied. In order to solve the problem, we introduce and study a new mathematical concept – generalized positive definite kernel (GPDK). With the help of GPDK and its properties, we obtain an efficient criterion of the polynomial stability for evolution equations with such a general but more complicated and useful memory. Moreover, in contrast to existing positive definite kernels, GPDK allows us to directly express the decay rate of the related kernel.

Citation: Kun-Peng Jin, Jin Liang, Ti-Jun Xiao. Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3141-3166. doi: 10.3934/dcdss.2021077
References:
[1]

S. Acosta and B. Palacios, Thermoacoustic tomography for an integro-differential wave equation modeling attenuation, J. Differential Equations, 264 (2010), 1984-2010.  doi: 10.1016/j.jde.2017.10.012.

[2]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[3]

P. Cannarsa and D. Sforza, Semilinear integrodifferential equations of hyperbolic type: Existence in the large, Mediterr. J. Math., 1 (2004), 151-174.  doi: 10.1007/s00009-004-0009-3.

[4]

P. Cannarsa and D. Sforza, Integro-differential equations of hyperbolic type with positive definite kernels, J. Differential Equations, 250 (2011), 4289-4335.  doi: 10.1016/j.jde.2011.03.005.

[5]

M. M. CavalcantiF. R. Dias SilvaV. N. Domingos Cavalcanti and A. Vicente, Stability for the mixed problem involving the wave equation, with localized damping, in unbounded domains with finite measure, SIAM J. Control Optim., 56 (2018), 2802-2834.  doi: 10.1137/16M1100514.

[6]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. Jorge Silva and A. Y. de Souza Franco, Exponential stability for the wave model with localized memory in a past history framework, J. Differential Equations, 264 (2018), 6535-6584.  doi: 10.1016/j.jde.2018.01.044.

[7]

M. M. CavalcantiI. Lasiecka and D. Toundykov, Wave equation with damping affecting only a subset of static Wentzell boundary is uniformly stable, Trans. Amer. Math. Soc., 364 (2012), 5693-5713.  doi: 10.1090/S0002-9947-2012-05583-8.

[8]

C. M. Dafermos and J. A. Nokel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations, 4 (1979), 219-278.  doi: 10.1080/03605307908820094.

[9]

B. de Andrade and A. Viana, Abstract Volterra integrodifferential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175.  doi: 10.1007/s00208-016-1469-z.

[10]

V. GeorgievB. Rubino and R. Sampalmieri, Global existence for elastic waves with memory, Arch. Ration. Mech. Anal., 176 (2005), 303-330.  doi: 10.1007/s00205-004-0345-2.

[11]

G. Gripenberg, S. O. Londen and O. J. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl., vol. 34, Cambridge Univ. Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[12]

W. J. Hrusa, Global existence and asymptotic stability for a semilinear Volterra equation with large initial data, SIAM J. Math. Anal., 16 (1985), 110-134.  doi: 10.1137/0516007.

[13]

K.-P. JinJ. Liang and T.-J. Xiao, Coupled second order evolution equations with fading memory: Optimal energy decay rate, J. Differential Equations, 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018.

[14]

K.-P. JinJ. Liang and T.-J. Xiao, Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings, J. Differential Equations, 266 (2019), 7230-7263.  doi: 10.1016/j.jde.2018.11.031.

[15]

K.-P. JinJ. Liang and T.-J. Xiao, Asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory, J. Math. Anal. Appl., 475 (2019), 554-575.  doi: 10.1016/j.jmaa.2019.02.055.

[16]

S. Kawashima, Global solutions to the equation of viscoelasticity with fading memory, J. Differential Equations, 101 (1993), 388-420.  doi: 10.1006/jdeq.1993.1017.

[17]

I. Lasiecka, S. A. Messaoudi and M. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys., 54 (2013), 031504. doi: 10.1063/1.4793988.

[18]

C. Li, J. Liang and T.-J. Xiao, Long-term dynamical behavior of the wave model with locally distributed frictional and viscoelastic damping, Commun. Nonlinear Sci. Numer. Simulat., 92 (2021), 105472, 22 pp. doi: 10.1016/j.cnsns.2020.105472.

[19]

S.-O. Londen and W. M. Ruess, Linearized stability for nonlinear Volterra equations, J. Evol. Equ., 17 (2017), 473-483.  doi: 10.1007/s00028-016-0381-z.

[20]

P. Loreti and D. Sforza, A Semilinear Integro-Differential Equation: Global Existence and Hidden Regularity, in Trends in Control Theory and Partial Differential Equations (eds. F. Alabau-Boussouira, F. Ancona, A. Porretta and C. Sinestrari), Springer INdAM Series, vol 32. Springer, Cham., 2019.

[21]

R. C. Maccamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc., 164 (1972), 1-37.  doi: 10.1090/S0002-9947-1972-0293355-X.

[22]

J. E. Mu${\rm\tilde{n}}$oz Rivera and E. C. Lapa, Decay rates of solutions of an anisotropic inhomogeneous $n$-dimensional viscoelastic equation with polynomial decaying kernels, Comm. Math. Phys., 177 (1996), 583-602.  doi: 10.1007/BF02099539.

[23]

J. E. Mu${\rm\tilde{n}}$oz Rivera and H. D. Fern${\rm\acute{a}}$ndez Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.  doi: 10.1016/j.jmaa.2007.07.012.

[24]

M. Nakao, $L^{p}$ estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains, Math. Ann., 320 (2001), 11-31.  doi: 10.1007/PL00004463.

[25]

S. Nicaise and C. Pignotti, Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029.

[26]

S. NicaiseC. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 3 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[27]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[28]

J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations, Adv. Math., 22 (1976), 278-304.  doi: 10.1016/0001-8708(76)90096-7.

[29]

M. Okada and S. Kawashima, Global solutions to the equation of thermoelasticity with fading memory, J. Differential Equations, 263 (2017), 338-364.  doi: 10.1016/j.jde.2017.02.037.

[30]

O. J. Staffans, On a nonlinear hyperbolic Volterra equation, SIAM J. Math. Anal., 11 (1980), 793-812.  doi: 10.1137/0511071.

[31]

J. S. W. Wong, Positive definite functions and Volterra integral equations, Bull. Amer. Math. Soc., 80 (1974), 679-682.  doi: 10.1090/S0002-9904-1974-13546-9.

[32]

T.-J. Xiao and J. Liang, Coupled second order semilinear evolution equations indirectly damped via memory effects, J. Differential Equations, 254 (2013), 2128-2157.  doi: 10.1016/j.jde.2012.11.019.

[33]

H. Zhan and Z. Feng, Stability of hyperbolic-parabolic mixed type equations with partial boundary condition, J. Differential Equations, 264 (2018), 7384-7411.  doi: 10.1016/j.jde.2018.02.019.

[34]

H. Zhan and Z. Feng, Stability of hyperbolic-parabolic mixed type equations, Dyn. Partial Differ. Equ., 16 (2019), 253-272.  doi: 10.4310/DPDE.2019.v16.n3.a2.

show all references

References:
[1]

S. Acosta and B. Palacios, Thermoacoustic tomography for an integro-differential wave equation modeling attenuation, J. Differential Equations, 264 (2010), 1984-2010.  doi: 10.1016/j.jde.2017.10.012.

[2]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[3]

P. Cannarsa and D. Sforza, Semilinear integrodifferential equations of hyperbolic type: Existence in the large, Mediterr. J. Math., 1 (2004), 151-174.  doi: 10.1007/s00009-004-0009-3.

[4]

P. Cannarsa and D. Sforza, Integro-differential equations of hyperbolic type with positive definite kernels, J. Differential Equations, 250 (2011), 4289-4335.  doi: 10.1016/j.jde.2011.03.005.

[5]

M. M. CavalcantiF. R. Dias SilvaV. N. Domingos Cavalcanti and A. Vicente, Stability for the mixed problem involving the wave equation, with localized damping, in unbounded domains with finite measure, SIAM J. Control Optim., 56 (2018), 2802-2834.  doi: 10.1137/16M1100514.

[6]

M. M. CavalcantiV. N. Domingos CavalcantiM. A. Jorge Silva and A. Y. de Souza Franco, Exponential stability for the wave model with localized memory in a past history framework, J. Differential Equations, 264 (2018), 6535-6584.  doi: 10.1016/j.jde.2018.01.044.

[7]

M. M. CavalcantiI. Lasiecka and D. Toundykov, Wave equation with damping affecting only a subset of static Wentzell boundary is uniformly stable, Trans. Amer. Math. Soc., 364 (2012), 5693-5713.  doi: 10.1090/S0002-9947-2012-05583-8.

[8]

C. M. Dafermos and J. A. Nokel, Energy methods for nonlinear hyperbolic Volterra integrodifferential equations, Comm. Partial Differential Equations, 4 (1979), 219-278.  doi: 10.1080/03605307908820094.

[9]

B. de Andrade and A. Viana, Abstract Volterra integrodifferential equations with applications to parabolic models with memory, Math. Ann., 369 (2017), 1131-1175.  doi: 10.1007/s00208-016-1469-z.

[10]

V. GeorgievB. Rubino and R. Sampalmieri, Global existence for elastic waves with memory, Arch. Ration. Mech. Anal., 176 (2005), 303-330.  doi: 10.1007/s00205-004-0345-2.

[11]

G. Gripenberg, S. O. Londen and O. J. Staffans, Volterra Integral and Functional Equations, Encyclopedia Math. Appl., vol. 34, Cambridge Univ. Press, Cambridge, 1990. doi: 10.1017/CBO9780511662805.

[12]

W. J. Hrusa, Global existence and asymptotic stability for a semilinear Volterra equation with large initial data, SIAM J. Math. Anal., 16 (1985), 110-134.  doi: 10.1137/0516007.

[13]

K.-P. JinJ. Liang and T.-J. Xiao, Coupled second order evolution equations with fading memory: Optimal energy decay rate, J. Differential Equations, 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018.

[14]

K.-P. JinJ. Liang and T.-J. Xiao, Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings, J. Differential Equations, 266 (2019), 7230-7263.  doi: 10.1016/j.jde.2018.11.031.

[15]

K.-P. JinJ. Liang and T.-J. Xiao, Asymptotic behavior for coupled systems of second order abstract evolution equations with one infinite memory, J. Math. Anal. Appl., 475 (2019), 554-575.  doi: 10.1016/j.jmaa.2019.02.055.

[16]

S. Kawashima, Global solutions to the equation of viscoelasticity with fading memory, J. Differential Equations, 101 (1993), 388-420.  doi: 10.1006/jdeq.1993.1017.

[17]

I. Lasiecka, S. A. Messaoudi and M. Mustafa, Note on intrinsic decay rates for abstract wave equations with memory, J. Math. Phys., 54 (2013), 031504. doi: 10.1063/1.4793988.

[18]

C. Li, J. Liang and T.-J. Xiao, Long-term dynamical behavior of the wave model with locally distributed frictional and viscoelastic damping, Commun. Nonlinear Sci. Numer. Simulat., 92 (2021), 105472, 22 pp. doi: 10.1016/j.cnsns.2020.105472.

[19]

S.-O. Londen and W. M. Ruess, Linearized stability for nonlinear Volterra equations, J. Evol. Equ., 17 (2017), 473-483.  doi: 10.1007/s00028-016-0381-z.

[20]

P. Loreti and D. Sforza, A Semilinear Integro-Differential Equation: Global Existence and Hidden Regularity, in Trends in Control Theory and Partial Differential Equations (eds. F. Alabau-Boussouira, F. Ancona, A. Porretta and C. Sinestrari), Springer INdAM Series, vol 32. Springer, Cham., 2019.

[21]

R. C. Maccamy and J. S. W. Wong, Stability theorems for some functional equations, Trans. Amer. Math. Soc., 164 (1972), 1-37.  doi: 10.1090/S0002-9947-1972-0293355-X.

[22]

J. E. Mu${\rm\tilde{n}}$oz Rivera and E. C. Lapa, Decay rates of solutions of an anisotropic inhomogeneous $n$-dimensional viscoelastic equation with polynomial decaying kernels, Comm. Math. Phys., 177 (1996), 583-602.  doi: 10.1007/BF02099539.

[23]

J. E. Mu${\rm\tilde{n}}$oz Rivera and H. D. Fern${\rm\acute{a}}$ndez Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.  doi: 10.1016/j.jmaa.2007.07.012.

[24]

M. Nakao, $L^{p}$ estimates for the linear wave equation and global existence for semilinear wave equations in exterior domains, Math. Ann., 320 (2001), 11-31.  doi: 10.1007/PL00004463.

[25]

S. Nicaise and C. Pignotti, Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029.

[26]

S. NicaiseC. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 3 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[27]

S. NicaiseJ. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[28]

J. A. Nohel and D. F. Shea, Frequency domain methods for Volterra equations, Adv. Math., 22 (1976), 278-304.  doi: 10.1016/0001-8708(76)90096-7.

[29]

M. Okada and S. Kawashima, Global solutions to the equation of thermoelasticity with fading memory, J. Differential Equations, 263 (2017), 338-364.  doi: 10.1016/j.jde.2017.02.037.

[30]

O. J. Staffans, On a nonlinear hyperbolic Volterra equation, SIAM J. Math. Anal., 11 (1980), 793-812.  doi: 10.1137/0511071.

[31]

J. S. W. Wong, Positive definite functions and Volterra integral equations, Bull. Amer. Math. Soc., 80 (1974), 679-682.  doi: 10.1090/S0002-9904-1974-13546-9.

[32]

T.-J. Xiao and J. Liang, Coupled second order semilinear evolution equations indirectly damped via memory effects, J. Differential Equations, 254 (2013), 2128-2157.  doi: 10.1016/j.jde.2012.11.019.

[33]

H. Zhan and Z. Feng, Stability of hyperbolic-parabolic mixed type equations with partial boundary condition, J. Differential Equations, 264 (2018), 7384-7411.  doi: 10.1016/j.jde.2018.02.019.

[34]

H. Zhan and Z. Feng, Stability of hyperbolic-parabolic mixed type equations, Dyn. Partial Differ. Equ., 16 (2019), 253-272.  doi: 10.4310/DPDE.2019.v16.n3.a2.

[1]

Narcisa Apreutesei, Nikolai Bessonov, Vitaly Volpert, Vitali Vougalter. Spatial structures and generalized travelling waves for an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 537-557. doi: 10.3934/dcdsb.2010.13.537

[2]

Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3637-3654. doi: 10.3934/dcdsb.2016114

[3]

Giuseppe Maria Coclite, Mario Michele Coclite. Positive solutions of an integro-differential equation in all space with singular nonlinear term. Discrete and Continuous Dynamical Systems, 2008, 22 (4) : 885-907. doi: 10.3934/dcds.2008.22.885

[4]

Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015

[5]

Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete and Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417

[6]

Xinjie Dai, Aiguo Xiao, Weiping Bu. Stochastic fractional integro-differential equations with weakly singular kernels: Well-posedness and Euler–Maruyama approximation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4231-4253. doi: 10.3934/dcdsb.2021225

[7]

Walter Allegretto, John R. Cannon, Yanping Lin. A parabolic integro-differential equation arising from thermoelastic contact. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 217-234. doi: 10.3934/dcds.1997.3.217

[8]

Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129

[9]

Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907

[10]

Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57

[11]

Abdelaziz Soufyane, Belkacem Said-Houari. The effect of the wave speeds and the frictional damping terms on the decay rate of the Bresse system. Evolution Equations and Control Theory, 2014, 3 (4) : 713-738. doi: 10.3934/eect.2014.3.713

[12]

Michel Chipot, Senoussi Guesmia. On a class of integro-differential problems. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1249-1262. doi: 10.3934/cpaa.2010.9.1249

[13]

Monica Conti, Lorenzo Liverani, Vittorino Pata. On the optimal decay rate of the weakly damped wave equation. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2022107

[14]

Yong-Kui Chang, Xiaojing Liu. Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability. Evolution Equations and Control Theory, 2020, 9 (3) : 845-863. doi: 10.3934/eect.2020036

[15]

Amelia Álvarez, José-Luis Bravo, Manuel Fernández. The number of limit cycles for generalized Abel equations with periodic coefficients of definite sign. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1493-1501. doi: 10.3934/cpaa.2009.8.1493

[16]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057

[17]

Nestor Guillen, Russell W. Schwab. Neumann homogenization via integro-differential operators. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3677-3703. doi: 10.3934/dcds.2016.36.3677

[18]

Paola Loreti, Daniela Sforza. Observability of $N$-dimensional integro-differential systems. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 745-757. doi: 10.3934/dcdss.2016026

[19]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021051

[20]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems on degenerate integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022025

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (172)
  • HTML views (130)
  • Cited by (0)

Other articles
by authors

[Back to Top]