May  2022, 15(5): 1307-1316. doi: 10.3934/dcdss.2021129

General and optimal decay for a quasilinear parabolic viscoelastic system

1. 

Department of Engineering Processes, University of Bejaia, Bejaia 06000, Algeria

2. 

Department of Mathematics, University of Sharjah, Sharjah, P.O. Box 27272, UAE

* Corresponding author: Abderrahmane Youkana

Received  June 2021 Revised  August 2021 Published  May 2022 Early access  October 2021

In this paper, we give a general decay rate for a quasilinear parabolic viscoelatic system under a general assumption on the relaxation functions satisfying $ g'(t) \leq - \xi(t) H(g(t)) $, where $ H $ is an increasing, convex function and $ \xi $ is a nonincreasing function. Precisely, we establish a general and optimal decay result for a large class of relaxation functions which improves and generalizes several stability results in the literature. In particular, our result extends an earlier one in the literature, namely, the case of the polynomial rates when $ H(t) = t^p, \ t\geq 0, \forall p>1 $, instead the parameter $ p \in [1, \frac{3}{2}[ $.

Citation: Abderrahmane Youkana, Salim A. Messaoudi. General and optimal decay for a quasilinear parabolic viscoelastic system. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1307-1316. doi: 10.3934/dcdss.2021129
References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[2]

S. Berrimi and S. A. Messaoudi, A decay result for a quasilinear parabolic system, Progr. Nonlinear Differential Equations Appl., 63 (2005), 43-50.  doi: 10.1007/3-7643-7384-9_5.

[3]

H. EnglernB. Kawohl and S. Luckhaus, Gradient estimates for solutions of parabolic equations and systems, J. Math. Anal. Appl., 147 (1990), 309-329.  doi: 10.1016/0022-247X(90)90350-O.

[4]

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

[5]

G. Liu and H. Chen, Global and blow-up of solutions for a quasilinear parabolic system with viscoelastic and source terms, Math. Methods Appl. Sci., 37 (2014), 148-156.  doi: 10.1002/mma.2792.

[6]

S. A. Messaoudi and B. Tellab, A general decay result in a quasilinear parabolic system with viscoelastic term, Appl. Math. Lett., 25 (2012), 443-447.  doi: 10.1016/j.aml.2011.09.033.

[7]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[8]

M. Nakao and Y. Ohara, Gradient estimates for a quasilinear parabolic equation of the mean curvature type, J. Math. Soc. Japan, 3 (1996), 455-466.  doi: 10.2969/jmsj/04830455.

[9]

M. Nakao and C. Chen, Global existence and gradient estimates for the quasilinear parabolic equations of $m$-Laplacian type with a nonlinear convection term, J. Differential Equations, 162 (2000), 224-250.  doi: 10.1006/jdeq.1999.3694.

[10]

J. A. Nohel, Nonlinear Volterra equations for the heat flow in materials with memory, in: Integral and Functional Differential Equations, pp. 3–82, Lecture Notes in Pure and Appl. Math., 67, Dekker, New York, 1981.

[11]

G. Da Prato and M. Iannelli, Existence and regularity for a class of integro-differential equations of parabolic type, J. Math. Anal. Appl., 112 (1985), 36-55.  doi: 10.1016/0022-247X(85)90275-6.

[12]

P. Pucci and J. Serrin, Asymptotic stability for nonlinear parabolic systems, in: Energy Methods in Continuum Mechanics, 66–74, Kluwer Acad. Publ., Dordrecht, 1996.

[13]

H.-M. Yin, On parabolic Volterra equations in several space dimensions, SIAM J. Math. Anal., 22 (1991), 1723-1737.  doi: 10.1137/0522106.

[14]

A. YoukanaS. A. Messaoudi and A. Guesmia, A general decay and optimal decay result in a heat system with a viscoelastic term, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 618-626.  doi: 10.1007/s10473-019-0223-5.

show all references

References:
[1]

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4757-2063-1.

[2]

S. Berrimi and S. A. Messaoudi, A decay result for a quasilinear parabolic system, Progr. Nonlinear Differential Equations Appl., 63 (2005), 43-50.  doi: 10.1007/3-7643-7384-9_5.

[3]

H. EnglernB. Kawohl and S. Luckhaus, Gradient estimates for solutions of parabolic equations and systems, J. Math. Anal. Appl., 147 (1990), 309-329.  doi: 10.1016/0022-247X(90)90350-O.

[4]

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

[5]

G. Liu and H. Chen, Global and blow-up of solutions for a quasilinear parabolic system with viscoelastic and source terms, Math. Methods Appl. Sci., 37 (2014), 148-156.  doi: 10.1002/mma.2792.

[6]

S. A. Messaoudi and B. Tellab, A general decay result in a quasilinear parabolic system with viscoelastic term, Appl. Math. Lett., 25 (2012), 443-447.  doi: 10.1016/j.aml.2011.09.033.

[7]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[8]

M. Nakao and Y. Ohara, Gradient estimates for a quasilinear parabolic equation of the mean curvature type, J. Math. Soc. Japan, 3 (1996), 455-466.  doi: 10.2969/jmsj/04830455.

[9]

M. Nakao and C. Chen, Global existence and gradient estimates for the quasilinear parabolic equations of $m$-Laplacian type with a nonlinear convection term, J. Differential Equations, 162 (2000), 224-250.  doi: 10.1006/jdeq.1999.3694.

[10]

J. A. Nohel, Nonlinear Volterra equations for the heat flow in materials with memory, in: Integral and Functional Differential Equations, pp. 3–82, Lecture Notes in Pure and Appl. Math., 67, Dekker, New York, 1981.

[11]

G. Da Prato and M. Iannelli, Existence and regularity for a class of integro-differential equations of parabolic type, J. Math. Anal. Appl., 112 (1985), 36-55.  doi: 10.1016/0022-247X(85)90275-6.

[12]

P. Pucci and J. Serrin, Asymptotic stability for nonlinear parabolic systems, in: Energy Methods in Continuum Mechanics, 66–74, Kluwer Acad. Publ., Dordrecht, 1996.

[13]

H.-M. Yin, On parabolic Volterra equations in several space dimensions, SIAM J. Math. Anal., 22 (1991), 1723-1737.  doi: 10.1137/0522106.

[14]

A. YoukanaS. A. Messaoudi and A. Guesmia, A general decay and optimal decay result in a heat system with a viscoelastic term, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 618-626.  doi: 10.1007/s10473-019-0223-5.

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