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Growth, decay and bifurcation of shock amplitudes under the type-II flux law. (English) Zbl 1132.35413
Summary: By replacing Fick’s diffusion law with Green and Nagdhi’s type-II flux law, a hyperbolic counterpart to the classical Fisher-KPP equation is obtained. In this article, an analytical study of this partial differential equation is presented with an emphasis on shock and related kinematic wave phenomena. First, an exact travelling wave solution (TWS) is derived and examined. Then, using singular surface theory, exact amplitude expressions for both shock and acceleration waves are obtained. In addition, the issue of shock stability is addressed and the limitations of the model are noted.
It is shown that discontinuity (i.e. shock) formation in the TWS occurs only when the propagation speed, which must exceed the characteristic speed, tends to the latter. It is also shown that the shock amplitude equation undergoes a transcritical bifurcation. Lastly, numerical simulations of acceleration waves in a simple model problem are presented.

MSC:
35L60 First-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35B45 A priori estimates in context of PDEs
35L67 Shocks and singularities for hyperbolic equations
35B32 Bifurcations in context of PDEs
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[1] Fu, Y.B. & Scott, N.H. 1990 One-dimensional shock waves in simple materials with memory. <i>Proc. R. Soc. A</i> <b>428</b>, 547–571, (doi:10.1098/rspa.1990.0047). · Zbl 0725.73024
[2] Fu, Y.B. & Scott, N.H. 1991 The transition from acceleration wave to shock wave. <i>Int. J. Eng. Sci.</i> <b>29</b>, 617–624, (doi:10.1016/0020-7225(91)90066-C).
[3] Green, A.E. & Naghdi, P.M. 1991 A re-examination of the basic postulates of thermomechanics. <i>Proc. R. Soc. A</i> <b>432</b>, 171–194, (doi:10.1098/rspa.1991.0012). · Zbl 0726.73004
[4] Green, A.E. & Naghdi, P.M. 1993 Thermoelasticity without energy dissipation. <i>J. Elasticity</i> <b>31</b>, 189–208. · Zbl 0784.73009
[5] Gurtin, M.E. & Pipkin, A.C. 1968 A general theory of heat conduction with finite wave speeds. <i>Arch. Rat. Mech. Anal.</i> <b>31</b>, 113–126. · Zbl 0164.12901
[6] Hale, J. & Koçak, H. 1991 Dynamics and bifurcations. New York, NY: Springer, ch. 1, 2.
[7] Harris, S. 2005 Diffusive logistic population growth with immigration. <i>Appl. Math. Lett.</i> <b>18</b>, 261–265, (doi:10.1016/j.aml.2003.03.009). · Zbl 1064.92043
[8] Jaisaardsuetrong, J. & Straughan, B. 2007 Thermal waves in a rigid heat conductor. <i>Phys. Lett. A</i> <b>366</b>, 433–436, (doi:10.1016/j.physleta.2007.02.058).
[9] Jordan, P.M. 2005 Growth and decay of shock and acceleration waves in a traffic flow model with relaxation. <i>Physica D</i> <b>207</b>, 220–229, (doi:10.1016/j.physd.2005.06.002). · Zbl 1078.35073
[10] Jordan, P.M. 2005 Growth and decay of acoustic acceleration waves in Darcy-type porous media. <i>Proc. R. Soc. A</i> <b>461</b>, 2749–2766, (doi:10.1098/rspa.2005.1477). · Zbl 1186.76680
[11] Jordan, P.M. & Puri, A. 2002 Qualitative results for solutions of the steady Fisher–KPP equation. <i>Appl. Math. Lett.</i> <b>15</b>, 239–250, (doi:10.1016/S0893-9659(01)00124-0).
[12] Kay, A.L., Sherratt, J.A. & McLeod, J.B. 2001 Comparison theorems and variable speed waves in a scalar reaction–diffusion equation. <i>Proc. R. Soc. Edin. A</i> <b>131</b>, 1133–1161. · Zbl 0996.35034
[13] Kolmogoroff, A., Petrovsky, I. & Piscounoff, N. 1937 Étude de l’équations de la diffusion avec croissance de la quantité de matière et son application a un problème biologique. <i>Bull. Univ. Moskou, Ser. Int. Sec. A</i> <b>1</b>, 1–25. · Zbl 0018.32106
[14] Lighthill, M.J. & Whitham, G.B. 1955 On kinematic waves II. A theory of traffic flow on long crowded roads. <i>Proc. R. Soc. A</i> <b>229</b>, 317–345, (doi:10.1098/rspa.1955.0089). · Zbl 0064.20906
[15] Logan, J.D. 1994 An introduction to nonlinear partial differential equations. New York, NY: Wiley, pp. 196–199. · Zbl 0834.35001
[16] Méndez, V. & Camacho, J. 1997 Dynamics and thermodynamics of delayed population growth. <i>Phys. Rev. E</i> <b>55</b>, 6476–6482, (doi:10.1103/PhysRevE.55.6476).
[17] Mickens, R.E. & Jordan, P.M. 2004 A positivity–preserving nonstandard finite difference scheme for the DWE. <i>Num. Meth. Partial Diff. Eqn</i> <b>20</b>, 639–649, (doi:10.1002/num.20003).
[18] Mickens, R.E. & Jordan, P.M. 2005 A new positivity-preserving nonstandard finite difference scheme for the DWE. <i>Num. Meth. Partial Diff. Eqn</i> <b>21</b>, 976–985, (doi:10.1002/num.20073).
[19] Morro, A. 2006 Jump relations and discontinuity waves in conductors with memory. <i>Math. Comp. Model.</i> <b>43</b>, 138–149, (doi:10.1016/j.mcm.2005.04.016). · Zbl 1124.74026
[20] Müller, I. & Ruggeri, T. 1993 Extended thermodynamics. <i>Springer tracts in natural philosophy</i> (ed. Truesdell, C.), vol. 37. New York, NY: Springer
[21] Murray, J.D. 1993 Mathematical biology. 2nd edn. Berlin, Germany: Springer. · Zbl 0779.92001
[22] Needham, D.J. & King, A.C. 2002 The evolution of travelling waves in the weakly hyperbolic generalized Fisher model. <i>Proc. R. Soc. A</i> <b>458</b>, 1055–1088, (doi:10.1098/rspa.2001.0902). · Zbl 1001.76114
[23] Newman, W.I. 1980 Some exact solutions to a non-linear diffusion problem in population genetics and combustion. <i>J. Theor. Biol.</i> <b>85</b>, 325–334, (doi:10.1016/0022-5193(80)90024-7).
[24] Niwa, H.S. 1998 Migration dynamics of fish schools in heterothermal environments. <i>J. Theor. Biol.</i> <b>193</b>, 215–231, (doi:10.1006/jtbi.1998.0675).
[25] Ockendon, H., Ockendon, J.R. & Falle, S.A.E.G. 2001 The Fanno model for turbulent compressible flow. <i>J. Fluid Mech.</i> <b>445</b>, 187–206. · Zbl 0993.76036
[26] Ostoja-Starzewski, M. & Trebicki, J. 1999 On the growth and decay of acceleration waves in random media. <i>Proc. R. Soc. A</i> <b>455</b>, 2577–2614, (doi:10.1098/rspa.1999.0446). · Zbl 0941.74027
[27] Pierce, A.D. 1989 <i>Acoustics: an introduction to its physical principles and applications</i> pp. 589–591, New York, NY: Acoustical Society of America
[28] Quintanilla, R. & Straughan, B. 2004 A note on discontinuity waves in type III thermoelasticity. <i>Proc. R. Soc. A</i> <b>460</b>, 1169–1175, (doi:10.1098/rspa.2003.1131). · Zbl 1070.74024
[29] Rosen, G. 1966 Solutions of a certain nonlinear wave equation. <i>J. Math. Phys.</i> <b>45</b>, 235–265. · Zbl 0168.08103
[30] Sherratt, J.A. & Marchant, B.P. 1996 Algebraic decay and variable speeds in wavefront solutions of a scalar reaction–diffusion equation. <i>IMA J. Appl. Math.</i> <b>56</b>, 289–302. · Zbl 0858.35062
[31] Whitham, G.B. 1974 Linear and nonlinear waves. New York, NY: Wiley, pp. 26–30. · Zbl 0373.76001
[32] Xin, J. 2000 Front propagation in heterogeneous media. <i>SIAM Rev.</i> <b>42</b>, 161–230, (doi:10.1137/S0036144599364296).
[33] Aronson, D.G. & Weinberger, H.F. 1975 Nonlinear diffusion in population genetics, combustion, and nerve propagation. <i>Lecture Notes in Mathematics</i><i>Partial differential equations and related topics</i> (ed. Goldstein, J.A.), vol. 446. pp. 5–49, Berlin, Germany: Springer · Zbl 0325.35050
[34] Bland, D.R. 1988 Wave theory and applications. Oxford, UK: Oxford University Press. · Zbl 0648.73015
[35] Chandrasekharaiah, D.S. 1998 Hyperbolic thermoelasticity: a review of recent literature. <i>Appl. Mech. Rev.</i> <b>51</b>, 705–729.
[36] Chen, P.J. 1973 Growth and decay of waves in solids. <i>Handbuch der physik</i> (eds. Flügge, S. & Truesdell, C.), vol. VIa/3. pp. 303–402, Berlin, Germany: Springer
[37] Christov, C.I. & Jordan, P.M. 2005 Heat conduction paradox involving second-sound propagation in moving media. <i>Phys. Rev. Lett.</i> <b>94</b>, 154301. (doi:10.1103/PhysRevLett.94.154301).
[38] Christov, I., Christov, C.I. & Jordan, P.M. 2007 Cumulative nonlinear effects in acoustic wave propagation. <i>CMES: Comp. Model. Eng. Sci.</i> <b>17</b>, 47–54. · Zbl 1184.76847
[39] Debnath, L. 2004 Nonlinear partial differential equations for scientists and engineers. 2nd edn. Boston, MA: Birkhäuser. · Zbl 1242.35001
[40] Dodd, R.K., Eilbeck, J.C., Gibbon, J.D. & Morris, H.C. 1982 Solitons and nonlinear wave equations. London, UK: Academic Press, pp. 30–31. · Zbl 0496.35001
[41] Dolak, Y. & Hillen, T. 2003 [Erratum: ’Cattaneo models for chemosensitive movement: numerical solution and pattern formation’ <i>J. Math. Biol.</i> <b>46</b>, 153–170 (2003)]. <i>J. Math. Biol.</i> <b>46</b>, 461–478, (doi:10.1007/s00285-003-0221-y). · Zbl 1020.92004
[42] Dreyer, W. & Struchtrup, H. 1993 Heat pulse experiments revisited. <i>Cont. Mech. Thermodyn.</i> <b>5</b>, 3–50, (doi:10.1007/BF01135371).
[43] Ebert, U. & van Saarloos, W. 2000 Front propagation into unstable states: universal algebraic convergence towards uniformly translating pulled fronts. <i>Physica D</i> <b>146</b>, 1–99, (doi:10.1016/S0167-2789(00)00068-3). · Zbl 0984.35030
[44] Fisher, R.A. 1937 The wave of advance of advantageous genes. <i>Ann. Eugen.</i> <b>7</b>, 355–369.
[45] Franchi, F. & Straughan, B. 1994 Continuous dependence on the relaxation time and modelling, and unbounded growth, in theories of heat conduction with finite propagation speeds. <i>J. Math. Anal. Appl.</i> <b>185</b>, 726–746, (doi:10.1006/jmaa.1994.1279). · Zbl 0806.35009
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