Messaoudi, Salim A.; Tellab, Brahim A general decay result in a quasilinear parabolic system with viscoelastic term. (English) Zbl 1244.45004 Appl. Math. Lett. 25, No. 3, 443-447 (2012). Summary: We consider a quasilinear parabolic system of the form \[ A(t)|u_t|^{m-2}u_t-\Delta u+\int_{0}^{t}g(t-s)\Delta u(x,s)ds=0, \] for \(m\geq ~2,A (t)\) a bounded and positive definite matrix, and \(g\) a continuously differentiable decaying function and establish a general decay result from which the usual exponential and polynomial decay results are only special cases. Cited in 1 ReviewCited in 11 Documents MSC: 45K05 Integro-partial differential equations Keywords:heat conduction in materials with memory; exponential decay; quasilinear parabolic system; polynomial decay PDFBibTeX XMLCite \textit{S. A. Messaoudi} and \textit{B. Tellab}, Appl. Math. Lett. 25, No. 3, 443--447 (2012; Zbl 1244.45004) Full Text: DOI References: [1] Nohel, J. A., Nonlinear Volterra equations for the heat flow in materials with memory, (Integral and Functional Differential Equations. Integral and Functional Differential Equations, Lecture Notes in Pure and Applied Mathematics (1981), Marcel Dekker Inc.) · Zbl 0509.73042 [2] Da Prato, G.; Iannelli, M., Existence and regularity for a class of integro-differential equations of parabolic type, J. Math. Anal. Appl., 112, 36-55 (1985) · Zbl 0583.45009 [3] Yin, H. M., On parabolic Volterra equations in several space dimensions, SIAM J. Math. Anal., 22, 1723-1737 (1991) · Zbl 0745.45003 [4] Levine, H., Some nonexistence and instability theorems for solutions of formally parabolic equations of the form \(P u_t = - A u + F(u)\), Arch Ration. Mech. Anal., 51, 371-386 (1973) · Zbl 0278.35052 [5] Kalantarov, V. K.; Ladyzhenskaya, O. A., The occurrence of collapse for quasilinear equations of parabolic and hyperbolic type, J. Soviet Math., 10, 53-70 (1978) · Zbl 0388.35039 [6] Levine, H.; Park, S.; Serrin, J., Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type, J. Differential Equations, 142, 212-229 (1998) · Zbl 0891.35062 [7] Messaoudi, S. A., A note on blow up of solutions of a quasilinear heat equation with vanishing initial energy, J. Math. Anal. Appl., 273, 243-247 (2002) · Zbl 1121.35323 [8] Nakao, M.; Ohara, Y., Gradient estimates for a quasilinear parabolic equation of the mean curvature type, J. Math. Soc. Japan, 48 # 3, 455-466 (1996) · Zbl 1076.35523 [9] Nakao, M.; Chen, C., Global existence and gradient estimates for the quasilinear parabolic equations of \(m\)-Laplacian type with a nonlinear convection term, J. Differential Equations, 162, 224-250 (2000) · Zbl 0959.35103 [10] Englern, H.; Kawohl, B.; Luckhaus, S., Gradient estimates for solutions of parabolic equations and systems, J. Math. Anal. Appl., 147, 309-329 (1990) · Zbl 0708.35018 [11] Pucci, P.; Serrin, J., Asymptotic stability for nonlinear parabolic systems, (Energy Methods in Continuum Mechanics (1996), Kluwer Acad. Publ.: Kluwer Acad. Publ. Dordrecht) · Zbl 1011.35055 [12] Berrimi, S.; Messaoudi, S. A., A decay result for a quasilinear parabolic system, Progr. Nonlinear Differential Equations Appl., 53, 43-50 (2005) · Zbl 1082.35029 [13] Martinez, P., A new method to decay rate estimates for dissipative systems, ESAIM Control. Optim. Calc. Var., 4, 419-444 (1999) · Zbl 0923.35027 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.