Front propagation in heterogeneous media

SIAM Review

Volumn 42, Issue 2, 2000, Pages 161-230

  Xin, Jack ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

HEURISTIC METHODS; MATHEMATICAL MODELS; STATISTICAL METHODS;

ASYMPTOTIC EXPANSIONS; FRONT PROPAGATION; HETEROGENEOUS MEDIA; HOMOGENIZATION; REACTION-DIFFUSION-ADVECTION EQUATIONS;

MATHEMATICAL TECHNIQUES;

EID: 0034206088     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036144599364296     Document Type: Article

References (166)

1. D. G. ARONSON AND H. F. WEINBERGER, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978), pp. 33-76.
2. D. G. ARONSON AND H. F. WEINBERGER, Nonlinear diffusion in population genetics, combustion, and nerve propagation. Lecture Notes in Math. 446, Springer-Verlag, Berlin, 1975, pp. 5-49.
3. W. T. ASHURST AND G. I. SIVASHINSKY, On the flame propagation through periodic flow fields, Combustion Sei. Technol., 80 (1991), pp. 159-164.
4. M. AVELLANEDA AND A. J. MAJDA, Mathematical models with exact renormalization for turbulent transport, Comm. Math. Phys., 131 (1990), pp. 381-429.
5. M. AVELLANEDA AND A. J. MAJDA, Mathematical models with exact renormalization for turbulent transport II: Fractal interfaces, non-Gaussian statistics and the sweeping effect, Comm. Math. Phys., 146 (1992), pp. 139-204.
6. A.-L. BARABÁSI AND H. E. STANLEY, Fractal Concepts in Surface Growth, Cambridge University Press, Cambridge, UK, 1995.
7. G. I. BARENBLATT, Y. B. ZELDOVICH, AND A. G. ISTRANOV, On the diffusional-thermal stability of a laminar flame, Zh. Prikl. Mekh. Tekh. Fiz., 4 (1962), pp. 21-26.
8. G. BARLES, L. C. EVANS, AND P. E. SOUGANIDIS, Wavefront propagation for reaction-diffusion systems of PDE, Duke Math. J., 61 (1990). pp. 835-859.
9. G. K. BATCHELOR, The Theory of Homogeneous Turbulence, Cambridge Science Classics, Cambridge University Press, Cambridge, UK, 1982.
10. R. D. BENGURIA AND M. C. DEPASSIER, Validity of the linear speed selection mechanism for fronts of the nonlinear diffusion equation, Phys. Rev. Lett., 73 (1994), pp. 2272-2274.
11. R. D. BENGURIA AND M. C. DEPASSIER, Variational characterization of the speed of propagation of fronts for the nonlinear diffusion equation, Comm. Math. Phys., 175 (1996), pp. 221-227.
12. R. D. BENGURIA AND M. C. DEPASSIER, Speed of fronts of the reaction-diffusion equation, Phys. Rev. Lett., 77 (1996), pp. 1171-1173.
13. A. BENSOUSSAN, J.-L. LIONS, AND G. PAPANICOLAOU, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl. 5, North-Holland, Amsterdam, 1978.
14. H. BERESTYCKI AND B. LARROUTUROU, Quelque aspects mathématiques de la propagation des flames prémélangée, Pitman Res. Notes Math. Ser. 220, H. Brezis and J.-L. Lions, eds., 1991, p. 65.
15. H. BERESTYCKI, B. LARROUTUROU, AND P.-L. LIONS, Multi-dimensional travelling-wave solutions of a flame propagation model, Arch. Rational Mech. Anal., 111 (1990), pp. 33-49.
16. H. BERESTYCKI, B. LARROUTUROU, AND J.-M. ROQUEJOFFRE, Stability of traveling fronts in a model for flame propagation, part 1: Linear stability, Arch. Rational Mech. Anal., 117 (1992), pp. 97-117.
17. H. BERESTYCKI, B. NICOLAENKO. AND B. SCHEURER, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal., 16 (1985), pp. 1207-1242.
18. H. BERESTYCKI AND L. NIRENBERG, Some qualitative properties of solutions of semilinear elliptic equations in cylindrical domains, in Analysis et cetera, P. Rabinowitz and E. Zehnder, eds., Academic Press, New York, 1990, pp. 115-164.
19. H. BERESTYCKI AND L. NIRENBERG, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat., 22 (1991), pp. 1-37.
20. H. BERESTYCKI AND L. NIRENBERG, Traveling fronts in cylinders, Ann. Inst. H. Poincaré Anal. Non Linéare, 9 (1992), pp. 497-572.
21. J. BILLINGHAM AND D. J. NEEDHAM, A note on the properties of a family of traveling wave solutions arising in cubic autocatalysis, Dynam. Stability Systems, 6 (1991), pp. 33-49.
22. J. BILLINGHAM AND D. J. NEEDHAM, The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates, I. Permanent form traveling waves, Philos. Trans. Roy. Soc. London Ser. A, 334 (1991), pp. 1-24.
23. J. BILLINGHAM AND D. J. NEEDHAM, The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates, II. The initial value problem with an immobilized or nearly immobilized autocatalyst., Philos. Trans. Roy. Soc. London Ser. A, 336 (1991), pp. 497-539.
24. J. BILLINGHAM AND D. J. NEEDHAM, The development of traveling waves in quadratic and cubic autocatalysis with unequal diffusion rates, III. Large time development in quadratic autocatalysis, Quart. Appl. Math., 1 (1992), pp. 343-372.
25. P. BILLINGSLEY, Convergence of Probability Measures, John Wiley, New York, 1968.
26. A. BONNET, B. LARROUTUROU, AND L. SAINSAULIEU, On the stability of multiple steady planar flames when the Lewis number is less than one, Phys. D, 69 (1993), pp. 345-352.
27. W. BOSMA AND S. VAN DER ZEE, Transport of reacting solute in a one-dimensional chemically heterogeneous porous media, Water Resources Research, 29 (1993), pp. 117-131.
28. W. BOSMA, S. VAN DER ZEE, AND C. J. VAN DUIJN, Plume development of a nonlinearly adsorbing solute in heterogeneous porous formations, Water Resources Research, 32 (1996), pp. 1569-1584.
29. M. BRAMSON, Convergence of Solutions of the Kolmogorov Equations to Traveling Waves, Mem. Amer. Math. Soc. 44, Providence, RI, 1983.
30. L. BREIMAN, Probability, Classics Appl. Math., SIAM, Philadelphia, 1992.
31. J. BRICMONT AND A. KUPIAINEN, Stability of moving fronts in the Ginzburg-Landau equations, Comm. Math. Phys., 159 (1994), pp. 287-318.
32. J. BRICMONT AND A. KUPIAINEN, Renormalizing partial differential equations, in Constructive Physics, Results in Field Theory, Statistical Mechanics and Condensed Matter Physics. Proceedings of the Conference, Springer, Berlin, 1995, pp. 83-115.
33. J. BRICMONT, A. KUPIAINEN, AND J. XIN, Global large time self-similarity of a thermal-diffusive combustion system with critical nonlinearity, J. Differential Equations, 130 (1996), pp. 9-35.
34. L.-Y. CHEN, N. GOLDENFELD, AND Y. OONO, Renormalization group theory and variational calculations for propagating fronts, Phys. Rev. E, 49B (1994), pp. 4502-4511.
35. L.-Y. CHEN, N. GOLDENFELD, Y. OONO, AND G. PAQUETTE, Selection, stability and renormalization, Phys. A, 204 (1994), pp. 111-133.
36. P. CLAVIN AND F. A. WILLIAMS, Theory of premixed-flame propagation in large-scale turbulence, J. Fluid Mech., 90 (1979), pp. 598-604.
37. n, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 23 (1996), pp. 625-642.
38. M. G. CRANDALL, H. ISHII, AND P.-L. LIONS, User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc., 27 (1992), pp. 1-67.
39. M. G. CRANDALL, L. C. EVANS. AND P.-L. LIONS, Some properties of viscosity solutions of Hamilton-Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), pp. 487-502.
40. M. C. CROSS AND P. C. HOHENBERG, Pattern formation out of equilibrium, Rev. Modern Phys., 65 (1993), pp. 851-1112.
41. C. N. DAWSON, C. J. VAN DUIJN, AND R. E. GRUNDY, Large time asymptotics in contaminant transport in porous media, SIAM J. Appl. Math., 56 (1996), pp. 965-993.
42. D. A. DAWSON AND E. PERKINS, Historical Processes, Mem. Amer. Math. Soc. 0065-9266, Providence, RI, 1991.
43. M. D. DONSKER AND S. R. S. VARADHAN, Asymptotics for the Wiener sausage, Comm. Pure Appl. Math., 28 (1975), pp. 525-565.
44. R. DURRETT, Probability: Theory and Examples, 2nd ed., Wadsworth and Brooks, Belmont, CA, 1995.
45. J.-P. ECKMANN AND C. E. WAYNE, The nonlinear stability of front solutions for parabolic pde's, Comm. Math. Phys., 161 (1994), pp. 323-334.
46. P. F. EMBID, A. J. MAJDA, AND P. E. SOUGANIDIS, Comparison of turbulent flame speeds from complete averaging and the G-equation, Phys. Fluids, 7 (1995), pp. 2052-2060.
47. P. F. EMBID, A. J. MAJDA, AND P. E. SOUGANIDIS, Effective geometric front dynamics for premixed turbulent combustion with separated velocity scales, Combustion Sci. Techn., 103 (1994), pp. 85-100.
48. L. C. EVANS, Partial Differential Equations, AMS, Providence, RI, 1998.
49. L. C. EVANS, Weak Convergence Methods for Nonlinear PDEs, NSF-CBMS Regional Conference Ser. Math. 74, Providence. RI, 1990.
50. L. C. EVANS, Periodic homogenization of certain nonlinear PDE's, Proc. Roy. Soc. Edinburgh Sect. A, 120 (1992), pp. 245-265.
51. L. C. EVANS AND P. E. SOUGANIDIS, A PDE approach to geometric optics for certain semilinear parabolic equations, Indiana Math. J., 38 (1989), pp. 141-172.
52. L. C. EVANS AND P. E. SOUGANIDIS, A PDE approach to certain large deviation problems for systems of parabolic equations, Ann. Inst. H. Poincaré Anal. Non Linéare, 6 (1989), Suppl., pp. 229-258.
53. L. C. EVANS AND P. E. SOUGANIDIS, Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations, Indiana Math. J., 33 (1984), pp. 773-797.
54. E. B. FABES AND D. W. STROOCK, A new proof of Moser's parabolic Harnack inequality using the old ideas of Nash, Arch. Rational Mech. Anal., 96 (1986), pp. 327-338.
55. F. FAMILY AND T. VICSEK, EDS., Dynamics of Fractal Surfaces, World Scientific, Singapore, 1991.
56. A. FANNJIANG AND G. PAPANICOLAOU, Diffusion in Turbulence, Probab. Theory Related Fields, 105 (1996), pp. 279-334.
57. A. FANNJIANG AND G. PAPANICOLAOU, Convection enhanced diffusion for random flows, J. Statist. Phys., 88 (1997), pp. 1033-1076.
58. G. FENNEMORE AND J. X. XIN, Wetting fronts in one dimensional periodically layered soils, SIAM J. Appl. Math., 58 (1998), pp. 387-427.
59. P. C. FIFE, Mathematical Aspects of Reacting and Diffusing Systems, Lecture Notes in Biomath. 28, Springer-Verlag, New York, 1979.
60. P. C. FIFE, Dynamics of Internal Layers and Diffusive Interfaces, CBMS-NSF Regional Conf. Ser. Appl. Math. 53, SIAM, Philadelphia, 1988.
61. P. C. FIFE AND J. B. MCLEOD, The approach of solutions of nonlinear diffusion equations to traveling front solutions, Arch. Rational Mech. Anal., 65 (1977), pp. 335-362.
62. R. A. FISHER, The wave of advance of advantageous genes, Ann. Eugenics, 7 (1937), pp. 355-369.
63. W. H. FLEMING AND H. M. SONER, Controlled Markov Processes and Viscosity Solutions, Appl. Math. 25, Springer-Verlag, New York, 1993.
64. S. FOCANT AND TH. GALLEY, Existence and stability of propagating fronts for an autocatalytic reaction-diffusion system, Phys. D, 120 (1998), pp. 346-368.
65. M. I. FREIDLIN, Propagation of a concentration wave in the presence of random motion associated with the growth of a substance, Soviet Math. Dokl., 20 (1979), pp. 503-507.
66. M. I. FREIDLIN, Limit theorems for large deviations and reaction-diffusion equations, Ann. Probab., 13 (1985), pp. 639-675.
67. M. I. FREIDLIN, Functional Integration and Partial Differential Equations, Ann. Math. Stud. 109, Princeton University Press, Princeton, NJ, 1985.
68. M. I. FREIDLIN AND T.-Y. LEE, Large deviation principle for the diffusion-transmutation processes and Dirichlet problem for PDE systems with small parameter, Probab. Theory Related Fields, 105 (1996), pp. 227-254.
69. M. I. FREIDLIN AND T.-Y. LEE, Wave front propagation and large deviations for diffusion-transmutation process, Probab. Theory Related Fields, 106 (1996), pp. 39-70.
70. M. I. FREIDLIN AND A. D. WENTZELL, Random Perturbations of Dynamical Systems, Springer-Verlag, New York, 1984.
71. G. FUSCO AND J. K. HALE, Stable equilibrium in a scalar parabolic equation with variable coefficients, SIAM J. Math. Anal., 16 (1995), pp. 1152-1164.
72. TH. GALLAY, Local stability of critical fronts in nonlinear parabolic pde's, Nonlinearity, 7 (1994), pp. 741-764.
73. R. GARDNER, Existence of multidimensional traveling wave solutions of an initial boundary value problem, J. Differential Equations, 61 (1986), pp. 335-379.
74. R. GARDNER, C. K. R. T. JONES, AND T. KAPITULA, Stability of traveling waves for non-convex scalar viscous conservation laws, Comm. Pure. Appl. Math., 46 (1993), pp. 505-526.
75. J. GÄRTNER AND M. I. FREIDLIN, On the propagation of concentration waves in periodic and random media, Soviet Math. Dokl., 20 (1979), pp. 1282-1286.
76. J. GÄRTNER, Bistable reaction-diffusion equations and excitable media, Math. Nachr., 112 (1983), pp. 125-152.
77. D. GiLBARG AND N. TRUDINGER, Elliptic Partial Differential Equations of Second Order, 2nd ed., Springer-Verlag, New York, 1983.
78. R. E. GOLDSTEIN, P. NELSON, T. POWERS, AND U. SEIFERT, Front propagation in the pearling instability of tubular vesicles, J. Physique II, 6 (1996), pp. 767-796.
79. J. GOODMAN, Stability of viscous shock profiles in several space dimensions, Trans. Amer. Math. Soc., 311 (1989), pp. 683-695.
80. R. E. GRUNDY, C. J. VAN DUIJN, AND C. N. DAWSON, Asymptotic profiles with finite mass in one-dimensional contaminant transport through porous media: Fast reaction case, Quart. J. Mech. Appl. Math., 47 (1994), pp. 69-106.
81. K. P. HADELER AND F. ROTHE, Traveling fronts in nonlinear diffusion equations, J. Math. Bio., 2 (1975), pp. 251-263.
82. B. G. HASLAM AND P. D. RONNEY, Fractal properties of propagating fronts m a strongly stirred fluid, Phys. Fluids, 7 (1995), pp. 1931-1937.
83. S. HEINZE, A Variational Approach to Traveling Waves, preprint, 1996.
84. S. HEINZE, Homogenization of Flame Fronts, preprint, 1996.
85. S. HEINZE, Wave Solutions to Reaction-Diffusion Systems in Perforated Domains, preprint, 1996.
86. D. HENRY, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, New York, 1981.
87. D. HORVATH, V. PETROV, S. SCOTT, AND K. SCHOWALTER, Instabilities in propagating reaction-diffusion fronts, J. Chem. Phys., 98 (1993), pp. 6332-6343.
88. W. HUDSON AND B. ZINNER, Existence of traveling waves for reaction diffusion equations of Fisher type in periodic media, in Boundary Problems for Functional Differential Equations, World Scientific, Singapore, 1995, pp. 187-199.
89. A. M. IL'IN AND O. A. OLEINIK, Behavior of the solutions of the Cauchy problem for certain quasilinear equations for unbounded increase of time, AMS Trans. Ser. 2, 42 (1964), pp. 19-23.
90. C. JONES, Stability of traveling wave solutions of the FitzHugh-Nagumo system, Trans. Amer. Math. Soc., 286 (1984), pp. 431-469.
91. G. JOULIN AND P. CLAVIN, Linear stability analysis of nonadiabatic flames: Diffusion-thermal model, Combust. and Flame, 35 (1979), pp. 139-153.
92. G. JOULIN AND G. I. SIVASHINSKY, Pockets in premixed flames and combustion-rate, Combustion Sci. Tech., 77 (1991), pp. 329-335.
93. JA. KANEL, Stabilization of solutions of the Cauchy problem for equations encountered in combustion theory, Mat. Sb., 59 (1962), pp. 245-288.
94. M. KARDAR, G. PARISI, AND Y.-C. ZHANG, Dynamic scaling of growing interfaces, Phys. Rev. Lett., 56 (1986), pp. 889-892.
95. T. KATO, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
96. J. B. KELLER, Genetic variability due to geographical inhomogeneity, J. Math. Biol., 20 (1984), pp. 223-230.
97. A. R. KERSTEIN, W. T. ASHURST, AND F. A. WILLIAMS, Field equations for interface propagation in an unsteady homogeneous flow field, Phys. Rev. A, 37 (1988), pp. 2728-2731.
98. A. R. KERSTEIN AND W. T. ASHURST, Propagation rate of growing interfaces in stirred fluids, Phys. Rev. Lett., 68 (1992), p. 934.
99. K. KIRCHGÄSSNER, On the nonlinear dynamics of traveling fronts, J. Differential Equations, 96 (1992), pp. 256-278.
100. A. N. KOLMOGOROV, I. G. PETROVSKY, AND N. S. PISKUNOV, Etude de l'eq́uation de la diffusion avec croissance de la quantité de matière et son application àun problème biologique, Moskow Univ. Math. Bull., 1 (1937), pp. 1-25.
101. R. H. KRAICHNAN, Small-scale structure of a scalar field convected by turbulence, Phys. Fluids, 11 (1968), pp. 945-963.
102. T.-Y. LEE AND F. TORCASO, Wave propagation in a lattice KPP equation in random media, Ann. Probab., 26 (1998), pp. 1179-1197.
103. C. D. LEVERMORE AND J. XIN, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, Part II, Comm. Partial Differential Equations, 17 (1992), pp. 1901-1924.
104. C. LI, Monotonicity and symmetry on bounded and unbounded domains, I & II, Comm. Partial Differential Equations, 16 (1991), pp. 491-526, 585-615.
105. P. L. LIONS, G. C. PAPANICOLAOU, AND S. VARADHAN, Homogenization of Hamilton-Jacobi Equations, unpublished.
106. A. J. MAJDA AND P. E. SOUGANIDIS, Large scale front dynamics for turbulent reaction-diffusion equations with separated velocity scales, Nonlinearity, 7 (1994), pp. 1-30.
107. A. J. MAJDA AND P. E. SOUGANIDIS, Bounds on enhanced turbulent flame speeds for combustion with fractal velocity fields. J. Statist. Phys., 83 (1996), pp. 933-954.
108. A. MALEVANETS, A. CARETA, AND R. KAPRAL, Biscale chaos in propagating front, Phys. Rev. E, 52 (1995), pp. 4724-4735.
109. S. MALHAM AND J. XIN, Global solutions to a reactive Boussinesq system with front data on an infinite domain, Comm. Math. Phys., 193 (1998), pp. 287-316.
110. J.-F. MALLORDY AND J.-M. ROQUEJOFFRE, A parabolic equation of the KPP type in higher dimensions, SIAM J. Math. Anal., 26 (1996), pp. 1-20.
111. M. MARION, Qualitative properties of a nonlinear system for laminar flames without ignition temperature, Nonlinear Anal., 11 (1983), pp. 1269-1292.
112. G. MATHERON AND G. DE MARSILY, Is transport in porous media always diffusive? A counterexample, Water Resources Research, 16 (1980), pp. 901-917.
113. G. DE MARSILY, Quantitative Hydrogeology - Groundwater Hydrology for Engineers, Academic Press, New York, 1986.
114. R. MARTIN AND M. PIERRE, Nonlinear reaction-diffusion systems, in Nonlinear Equations in the Applied Sciences, W. F. Ames and C. Rogers, eds., Academic Press, Boston, 1992.
115. H. MCKEAN, Application of Brownian motion to the equation of KPP, Comm. Pure Appl. Math., 28 (1975), pp. 323-331.
116. M. METCALF, J. MERKIN, AND S. SCOTT, Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis, Proc. Roy. Soc. London Ser. A, 447 (1994), pp. 155-174.
117. C. MUELLER AND E. PERKINS, The compact support property of solutions to the heat equations with noise, Probab. Theory Related Fields, 93 (1992), pp. 1-29.
118. C. MUELLER AND R. SOWERS, Random travelling waves for the KPP equations with noise, J. Funct. Anal., 128 (1995), pp. 439-498.
119. J. D. MURRAY, Mathematical Biology, 2nd ed., Lecture Notes in Biomath. 19, Springer-Verlag, New York, 1993.
120. J. NASH, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math., 80 (1958), pp. 931-954.
121. S. NII, Stability of the traveling multiple-front (multiple-back) wave solutions of the FitzHugh-Nagumo equations, SIAM J. Math. Anal., 28 (1997), pp. 1094-1112.
122. H. OSADA, Diffusion processes with generators of generalized divergence form, J. Math. Kyoto Univ., 27 (1987), pp. 597-619.
123. S. OSHER AND J. RALSTON, Stability of traveling waves in convective porous medium equation, Comm. Pure Appl. Math., 35 (1982), pp. 737-749.
124. G. C. PAQUETTE AND Y. OONO, Structural stability and selection of propagating fronts in semilinear parabolic partial differential equations, Phys. Rev. E, 49 (1994), pp. 2368-2388.
125. G. PAPANICOLAOU AND S. VARADHAN, Boundary value problems with rapidly oscillating random coefficients, Colloq. Math. Soc. János Bolyai 27, Random Fields, Esztergom, Hungary, 1979, North Holland, Amsterdam, 1982, pp. 835-873.
126. G. PAPANICOLAOU AND X. XIN, Reaction-diffusion fronts in periodically layered media, J. Stat. Phys., 63 (1991), pp. 915-931.
127. J. PAUWELUSSEN, Nerve impulse propagation in a branching nerve system: A simple model, Phys. D., 4 (1981), pp. 67-88.
128. N. PETERS, A spectral closure for premixed turbulent combustion in the flamelet regime, J. Fluid Mech., 242 (1992), pp. 611-629.
129. M. POSTEL AND J. XIN, A numerical study of fronts in random media using a reactive solute transport model, Comp. Geosci., 1 (1997), pp. 251-270.
130. M. PROTTER AND H. WEINBERGER, Maximum Principles in Differential Equations, Prentice-Hall, Englewood Cliffs, NJ, 1967.
131. N. PROVATAS, T. ALA-NISSILA, M. GRANT, K. ELDER, AND L. PICHÉ, Scaling, propagation and kinetic roughening of flame fronts in random media, J. Statist. Phys., 81 (1995), pp. 737-759.
132. L. F. RICHARDSON, Atmospheric diffusion shown on a distance-neighbour graph, Proc. Roy. Soc. London Ser. A, 110 (1926), pp. 709-737.
133. L. C. ROGERS AND D. WILLIAMS, Diffusion, Markov Processes and Martingales, 1, Foundations, John Wiley, New York, 1994.
134. P. RONNEY, Some Open Issues in Premixed Turbulent Combustion, Lecture Notes in Phys. 449, Springer-Verlag, Berlin, 1995, pp. 1-22.
135. P. D. RONNEY, B. G. HASLAM, AND N. O. RHYS, Front propagation rates in randomly stirred media, Phys. Rev. Lett., 74 (1995), pp. 3804-3807.
136. J.-M. ROQUEJOFFRE, Stability of traveling fronts in a model for flame propagation, part 2: Nonlinear stability, Arch. Rational Mech. Anal., 117 (1992), pp. 119-153.
137. J.-M. ROQUEJOFFRE, Convergence to traveling waves for solutions of a class of semilinear parabolic equations, J. Differential Equations, 108 (1996), pp. 262-295.
138. J.-M. ROQUEJOFFRE, Global stability of traveling waves and convergence towards stacked families of waves in monotone parabolic systems, SIAM J. Math. Anal., 27 (1996), pp. 1261-1269.
139. J. RUBINSTEIN, P. STERNBERG, AND J. KELLER, Fast Reaction, Slow Diffusion, and Curve Shortening, SIAM J. Appl. Math., 49 (1989), pp. 116-133.
140. B. SANDSTEDE, Stability of N-fronts bifurcating from a twisted heteroclinic loop and an application to the FitzHugh-Nagumo equation, SIAM. J. Math. Anal., 29 (1998), pp. 183-207.
141. D. H. SATTINGER, On the stability of waves of nonlinear parabolic systems, Adv. Math., 22 (1976), pp. 312-355.
142. N. SHIGESADA, K. KAWASAKI, AND E. TERAMOTO, Traveling periodic waves in heterogeneous environments, Theor. Population Biol., 30 (1986), pp. 143-160.
143. S. S. SHY, P. D. RONNEY, S. G. BUCKLEY, AND V. YAKHOT, 24th International Symposium on Combustion, Combustion Institute, Pittsburgh, PA, 1992, pp. 543-551.
144. G. I. SIVASHINSKY, Instabilities, pattern formation, and turbulence in flames, Ann. Rev. Fluid Mech, 15 (1983), pp. 179-199.
145. G. I. SIVASHINSKY, Cascade renormalization theory of turbulent flame speed, Combustion Sci. Tech, 62 (1988), pp. 77-96.
146. J. SMOLLER, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York, 1983.
147. K. UCHIYAMA, The behavior of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), pp. 453-508.
148. S. E. A. T. M. VAN DER ZEE AND W. H. VAN RIEMSDIJK, Transport of reactive solute in spatially variable soil systems, Water Resources Research, 23 (1987), pp. 2059-2069.
149. C. J. VAN DUIJN AND P. KNABNER, Solute transport, in porous media with equilibrium and nonequilibrium multiple sites adsorption, travelling waves. J. Reine Angew. Math., 415 (1991), pp. 1-49.
150. S. R. S. VARADHAN, Stochastic Processes. Courant Institute Lecture Notes, New York University, 1968.
151. S. R. S. VARADHAN, Large Deviations and Applications, CBMS-NSF Regional Conf. Ser. Appl. Math. 46, SIAM, Philadelphia, 1984.
152. S. R. S. VARADHAN, Stochastic differential equations - Large deviations, in Critical Phenomenon, Random Systems, Gauge Theories, K. Osterwalder and R. Stora, eds., Les Houches, 1984.
153. A. VOLPERT. V. VOLPERT, AND V. VOLPERT, Traveling Wave Solutions of Parabolic Systems, Transl. Math. Monographs 140. AMS. Providence, RI, 1994.
154. V. VOLPERT AND A. VOLPERT, Existence and stability of stationary solutions for a class of semilinear parabolic systems, Comm. Partial Differential Equations, 18 (1993), pp. 2051-2069.
155. J. WEHR AND J. XIN, White noise perturbation of the viscous shock fronts of the Burgers equation, Comm. Math. Phys., 181 (1996), pp. 183-203.
156. J. WEHR AND J. XIN, Front speed in the Burgers equation with a random flux. J. Statist. Phys., 88 (1997), pp. 843-871.
157. J. XIN, Existence and stability of travelling waves in periodic media governed by a bistable nonlinearity, J. Dynam. Differential Equations, 3 (1991), pp. 541-573.
158. J. X. XIN, Multidimensional stability of traveling waves in a bistable reaction-diffusion equation, Part I, Comm. Partial Differential Equations, 17 (1992), pp. 1889-1899.
159. J. X. XIN, Existence and uniqueness of travelling wave solutions in a reaction-diffusion equation with combustion nonlinearity. Indiana Math. J., 40 (1991), pp. 985-1008.
160. J. X. XIN, Existence of planar flame fronts in convective-diffusive periodic media, Arch. Rational Mech. Anal., 121 (1992), pp. 205-233.
161. J. X. XIN, Existence and nonexistence of traveling waves and reaction-diffusion front propagation in periodic media, J. Statist. Phys., 73 (1993), pp. 893-926.
162. J. X. XIN, Existence of multidimensional traveling waves in transport of reactive solutes through periodic porous media, Arch. Rational Mech. Anal., 128 (1994), pp. 75-103.
163. J. X. XIN AND J. ZHU, Quenching and propagation of bistable reaction-diffusion fronts in multidimensional periodic media, Phys. D, 81 (1995). pp. 94-110.
164. J. X. XIN, Stability of traveling waves in a solute transport equation, J. Differential Equations. 135 (1997), pp. 269-298.
165. V. YAKHOT, Propagation velocity of premixed turbulent flames, Combustion Sci. Tech., 60 (1988), pp. 191-214.
166. E. ZEIDLER, Nonlinear Functional Analysis, Springer-Verlag, New York. 1984.