doi: 10.3934/dcdss.2021129
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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 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 & Continuous Dynamical Systems - S, 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

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

[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.  Google Scholar

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