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 & Continuous Dynamical Systems - S, 2021, 14 (9) : 3141-3166. doi: 10.3934/dcdss.2021077
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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