Kirane, M.; Tatar, N.-e. Convergence rates for a reaction-diffusion system. (English) Zbl 0984.35080 Z. Anal. Anwend. 20, No. 2, 347-357 (2001). The authors consider a four-dimensional reaction-diffusion system defined on an \(n\)-dimensional bounded region with a smooth boundary and zero-flux boundary conditions. The model is similar to a SIR model of H. Hoshino [Differ. Integral Equ. 9, No. 4, 761-778 (1996; Zbl 0852.35023)] and for special parameter values describes the spread of FIV among susceptible/infective male/female cats as proposed by W. E. Fitzgibbon, M. Langlais, M. E. Parrott and G. F. Webb [Nonlinear Anal., Theory Methods Appl. 25, No. 9-10, 975-989 (1995; Zbl 0836.92019)]. The authors mention standard results on the existence and asymptotic behavior of solutions and obtain exponential convergence rates for a system with unbounded time-dependent coefficients. Reviewer: David S.Boukal (České Budějovice) Cited in 7 Documents MSC: 35K57 Reaction-diffusion equations 35B40 Asymptotic behavior of solutions to PDEs 92D30 Epidemiology 92B05 General biology and biomathematics Keywords:exponential convergence rates; unbounded time-dependent coefficients Citations:Zbl 0852.35023; Zbl 0836.92019 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Butler, G and T. Rogers: A generalization of a lemma of Bihari and applications to pointwise estimates for integral equations. J. Math. Anal. Appl. 33 (1971), 77 - 81. · Zbl 0209.42503 · doi:10.1016/0022-247X(71)90183-1 [2] Fitzgibbon, W. E., Langlais, M., Parrott, M. E. and G. F. Webb: A diffusive system with age dependence modeling FIV. Nonlin. Anal.: Theory, Meth. & Appl. 25 (1995), 975 -988. · Zbl 0836.92019 · doi:10.1016/0362-546X(95)00092-A [3] Haraux, A. and M. Kirane: Estimations C1 pour des probl‘emes paraboliques semilinéaires. Ann. Fac. Sci. Toulouse 5 (1983), 265 - 280. · Zbl 0531.35048 · doi:10.5802/afst.598 [4] Haraux, A. and A. Youkana: On a result of K. Masuda concerning reaction-diffusion equations. T\hat ohoku Math. J. 40 (1988), 159 - 163. · Zbl 0689.35041 · doi:10.2748/tmj/1178228084 [5] Henry, D.: Geometric Theory of Semilinear Parabolic Equations. Lect. Notes Math. 840 (1981). · Zbl 0456.35001 [6] Hoshino, H.: On the convergence properties of global solutions for some reaction diffusion systems under Neumann boundary conditions. Diff. Int. Equ. 9 (1996), 761 - 778. · Zbl 0852.35023 [7] Hoshino, H. and Y. Yamada: Solvability and smoothing effect for semilinear parabolic equations. Funkcialaj Ekvacioj 34 (1994), 475 - 494. · Zbl 0757.35033 [8] Hoshino, H. and Y. Yamada: Asymptotic behavior of global solutions for some reaction- diffusion systems. Nonlin. Anal.: Theory, Meth. & Appl. 23 (1994), 639 - 650. · Zbl 0811.35062 · doi:10.1016/0362-546X(94)90243-7 [9] Kirane, M. and N-E Tatar: Global existence and stability of some semilinear problems. Arch. Math. 36 (2000), 33 - 44. · Zbl 1048.34102 [10] Medved’, M.: A new approach to an analysis of Henry type integral inequalities and their Bihari type versions. J. Math. Anal. Appl. 214 (1997), 349 - 366. · Zbl 0893.26006 · doi:10.1006/jmaa.1997.5532 [11] Medved’, M.: Singular integral inequalities and stability of semilinear parabolic equations. Arch. Math. (Brno) 24 (1998), 183 - 190. · Zbl 0915.34057 [12] Michalski, M. W.: Derivatives of Noninteger Order and Their Applications (Diss. Math.). Warszawa: Polska Akad. Nauk, Inst. Mat. 1993. · Zbl 0880.26007 [13] Rothe, F.: Global Solutions of Reaction-Diffusion Systems. Lect. Notes Math. 1072 (1983). · Zbl 0546.35003 [14] Webb, G. F.: A reaction-diffusion model for a deterministic diffusive epidemic. J. Math. Anal. Appl. 84 (1981), 150 - 161. · Zbl 0484.92019 · doi:10.1016/0022-247X(81)90156-6 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.