The characterization of reaction-convection-diffusion processes by travelling waves

Journal of Differential Equations

Volumn 124, Issue 1, 1996, Pages 27-79

  Gilding, B H ^a   Kersner, R ^b  

^a UNIVERSITY OF TWENTE   (Netherlands)

^b INSTITUTE OF EXPERIMENTAL MEDICINE   (Hungary)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0029703999     PISSN: 00220396     EISSN: None     Source Type: Journal    
DOI: 10.1006/jdeq.1996.0002     Document Type: Article

References (173)

1. p with 0
2. 2. H. W. ALT AND S. LUCKHAUS, Quasilinear elliptic-parabolic differential equations, Math. Z. 183 (1983), 311-341.
3. 3. L. ALVAREZ, J. I. DIAZ, AND R. KERSNER, On the initial growth of the interfaces in nonlinear diffusion-convection processes, in "Nonlinear Diffusion Equations and Their Equilibrium States I" (W.-M. Ni, L. A. Peletier, and J. Serrin, Eds.), pp. 1-20, Springer-Verlag, New York, 1988.
4. 4. W. F. AMES, "Nonlinear Partial Differential Equations in Engineering," Academic Press, New York, 1965.
5. 5. W. F. AMES, "Nonlinear Partial Differential Equations in Engineering Volume II," Academic Press, New York, 1972.
6. 6. J. R. ANDERSON, Local existence and uniqueness of solutions of degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), 105-143.
7. 7. R. ARIS, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Volume I, The Theory of the Steady State," Clarendon Press, Oxford, 1975.
8. 8. R. ARIS, "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts, Volume II, Questions of Uniqueness, Stability and Transient Behaviour," Clarendon Press, Oxford, 1975.
9. 9. D. ARONSON, M. G. CRANDALL, AND L. A. PELETIER, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), 1001-1022.
10. 10. D. G. ARONSON, The porous medium equation, in "Nonlinear Diffusion Problems" (A. Fasano and M. Primicerio, Eds.), pp. 1-46, Lecture Notes in Mathematics, Vol. 1224, Springer-Verlag, Berlin, 1986.
11. 11. D. G. ARONSON, L. A. CAFFARELLI, AND J. L. VAZQUEZ, Interfaces with a corner point in one-dimensional porous medium flow, Comm. Pure Appl. Math. 38 (1985), 375-404.
12. 12. D. G. ARONSON AND H. F. WEINBERGER, Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation, in "Partial Differential Equations and Related Topics" (J. A. Goldstein, Ed.), pp. 5-49, Lecture Notes in Mathematics, Vol.446, Springer-Verlag, Berlin, 1975.
13. 13. G. I. BARENBLATT, On some unsteady motions of a liquid and gas in a porous medium, Prikl. Mat. Mekh. 16 (1952), 67-78.
14. 14. G. I. BLARENBLATT, Self-similarity: Similarity and intermediate asymptotic form, Radiophys. Quantum Electron. 19 (1976), 643-664. Translation of Izv. Vyssh. Uchebn. Zaved. Radiofiz. 19 (1976), 902-931.
15. 14. G. I. BLARENBLATT, Self-similarity: Similarity and intermediate asymptotic form, Radiophys. Quantum Electron. 19 (1976), 643-664. Translation of Izv. Vyssh. Uchebn. Zaved. Radiofiz. 19 (1976), 902-931.
16. 15. G. I. BRARENBLATT, "Similarity, Self-Similarity, and Intermediate Asymptotics," Consultants Bureau, New York, 1979.
17. 16. G. I. BARENBLATT, "Dimensional Analysis," Gordon and Breach Science Publishers, New York, 1987.
18. 17. G. I. BARENBLATT AND YA. B. ZEL'DOVICH, Intermediate asymptotics in mathematical physics, Russ. Math. Surv. 26 (1971), 45-61. Translation of Uspekhi Mat. Nauk 26 (1971), 115-129.
19. 17. G. I. BARENBLATT AND YA. B. ZEL'DOVICH, Intermediate asymptotics in mathematical physics, Russ. Math. Surv. 26 (1971), 45-61. Translation of Uspekhi Mat. Nauk 26 (1971), 115-129.
20. 18. G. I. BARENBLATT AND YA. B. ZEL'DOVICH, Self-similar solutions as intermediate asymptotics, Annu. Rev. Fluid Mech. 4 (1972), 285-312.
21. 19. J. BEAR, "Dynamics of Fluids in Porous Media," American Elsevier, New York, 1972.
22. 20. J. BEBERNES AND D. EBERLY, "Mathematical Problems from Combustion Theory," Springer-Verlag, New York, 1989.
23. 21. P. BÉNILAN, "Evolution Equations and Accretive Operators," Lecture Notes taken by S. Lenhart, University of Kentucky, 1981.
24. 22. P. BÉNILAN AND J. I. DIAZ, Comparison of solutions of nonlinear evolution problems with different nonlinear terms, Israel J. Math. 42 (1982), 241-257.
25. x + v, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), 919-922.
26. 24. P. W. BRIDGMAN, "Dimensional Analysis," Yale Univ. Press, New Haven, CT, 1931.
27. 25. J. BUCKMASTER, Viscous sheets advancing over dry beds, J. Fluid Mech. 81 (1977), 735-756.
28. 26. J. D. BUCKMASTER AND G. S. S. LUDFORD, "Theory of Laminar Flames," Cambridge Univ. Press, Cambridge, U.K., 1982.
29. 27. J. D. BUCKMASTER AND G. S. S. LUDFORD, "Lectures on Mathematical Combustion," SIAM, Philadelphia, PA, 1983.
30. 28. J. M. BURGERS, Application of a model system to illustrate some points of the statistical theory of free turbulence, Proc. Kon. Nederl. Akad. Wetensch. 43 (1940), 2-12.
31. 29. J. M. BURGERS, A mathematical model illustrating the theory of turbulence, Adv. Appl. Mech. 1 (1948), 171-199.
32. 30. J. M. BURGERS, "The Nonlinear Diffusion Equation," Reidel, Dordrecht, 1975.
33. 31. C. CHEN, The propagation of nonstationary filtration with absorption, Acta. Math. Appl. Sinica (English Ser.) 3 (1987), 342-350.
34. 32. X.-Y. CHEN, H. MATANO, AND M. MIMURA, Finite-point extinction and continuity of interfaces in a nonlinear diffusion equation with strong absorption, J. Reine Angew. Math. 459 (1995), 1-36.
35. 33. E. C. CHILDS, "An Introduction to the Physical Basis of Soil Water Phenomena," Wiley, London, 1969.
36. 34. E. COMPARINI, R. RICCI, AND J. L. VAZQUEZ, Asymptotic behavior of the solutions of a nonlinear Fokker-Planck equation with Dirichlet boundary conditions, J. Math. Anal. Appl. 175 (1993), 606-631.
37. 35. T. B. CONSTABLE AND N. T. S. EVANS, A method for measuring the spatial distribution of tissue oxygen removal rates, Respir. Physiol. 25 (1975), 175-190.
38. 36. C. CORTAZAR AND M. ELGUETA, Localization and boundedness of the solutions of the Neumann problem for a filtration equation, Nonlinear Anal. 13 (1989), 33-41.
39. 37. J. CRANK, "The Mathematics of Diffusion," Clarendon Press, Oxford, 1975.
40. 38. J. CRANK, "Free and Moving Boundary Problems," Clarendon Press, Oxford, 1984.
41. 39. J. CRANK AND R. S. GUPTA, A moving boundary problem arising from the diffusion of oxygen in absorbing tissue, J. Inst. Math. Appl. 10 (1972), 19-33.
42. 40. J. I. DIAZ AND R. KERSNER, Non existence d'une des frontières libres dans une équation dégénérée en théorie de la filtration, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 505-508.
43. 41. J. I. DIAZ AND R. KERSNER, On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium, J. Differential Equations 69 (1987), 368-403.
44. 42. J. I. DIAZ AND R. KERSNER, On the behaviour and cases of nonexistence of the free boundary in a semibounded porous medium, J. Math. Anal. Appl. 132 (1988), 281-289.
45. 43. E. DI BENEDETTO AND R. E. SHOWALTER, Implicit degenerate evolution equations and applications, SIAM J. Math. Anal. 12 (1981), 731-751.
46. 44. O. DIEKMANN AND N. M. TEMME (Eds.), "Nonlinear Diffusion Problems," Mathematisch Centrum, Amsterdam, 1976.
47. 45. P. G. DRAZIN, "Solitons," Cambridge Univ. Press, Cambridge, U.K., 1983.
48. 46. P. G. DRAZIN AND R. S. JOHNSON, "Solitons: An Introduction," Cambridge Univ. Press, Cambridge, U.K., 1989.
49. 47. C. J. VAN DUIJN AND J. M. DE GRAAF, Large time behaviour of solutions of the porous medium equation with convection, J. Differential Equations 84 (1990), 183-203.
50. 48. C. J. VAN DUIJN AND P. KNABNER, Solute transport through porous media with slow adsorption, in "Free Boundary Problems: Theory and Applications, Vol. I" (K. H. Hoffmann and J. Sprekels, Eds.), pp. 375-388, Longman, Harlow, U.K., 1990.
51. 49. G. DUVAUT, Résolution d'un problème de Stefan (Fusion d'un bloc de glace à zéro degré), C. R. Acad. Sci. Paris Sér. A 276 (1973), 1461-1463.
52. 50. C. M. ELLIOTT AND J. R. OCKENDON, "Weak and Variational Methods for Moving Boundary Problems," Pitman, Boston, 1982.
53. 51. L. C. EVANS AND B. F. KNERR, Instantaneous shrinking of the support of nonnegative solutions to certain nonlinear parabolic equations and variational inequalities, Illinois J. Math. 23 (1979), 153-166.
54. 52. P. C. FIFE, Asymptotic states for equations of reaction and diffusion, Bull. Amer. Math. Soc. (N. S.) 84 (1978), 693-726.
55. 53. P. C. FIFE, "Mathematical Aspects of Reacting and Diffusing Systems," Lecture Notes in Biomathematics, Vol. 28, Springer-Verlag, Berlin, 1979.
56. 54. P. C. FIFE AND J. B. MCLEOD, The approach of solutions of nonlinear diffusion equations to travelling wave solutions, Bull. Amer. Math. Soc. (N. S.) 81 (1975), 1076-1078.
57. 55. P. C. FIFE AND J. B. MCLEOD, The approach of solutions of nonlinear diffusion equations to travelling front solutions, Arch. Rational Mech. Anal. 65 (1977), 335-361.
58. 56. R. A. FISHER, The wave of advance of advantageous genes, Ann. Eugen. 7 (1937), 355-369.
59. 57. G. FRANCSICS, On the porous medium equations with lower order singular nonlinear terms, Acta Math. Hungar. 45 (1985), 425-436.
60. 58. A. FRIEDMAN, "Partial Differential Equations of Parabolic Type," Prentice-Hall, Englewood Cliffs, NJ, 1964.
61. 1+α, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 13 (1966), 109-124.
62. 60. V. A. GALAKTIONOV, Some properties of travelling waves in a medium with non-linear thermal conductivity and a source of heat, U.S.S.R. Comput. Math. Math. Phys. 21 (1981), 167-176. Translation of Zh. Vychisl. Mat. Mat. Fiz. 21 (1981), 980-989.
63. 60. V. A. GALAKTIONOV, Some properties of travelling waves in a medium with non-linear thermal conductivity and a source of heat, U.S.S.R. Comput. Math. Math. Phys. 21 (1981), 167-176. Translation of Zh. Vychisl. Mat. Mat. Fiz. 21 (1981), 980-989.
64. x + uβ, Differential Equations 21 (1985), 11-18. Translation of Differentsial'nye Urav. 21 (1985), 15-23.
65. x + uβ, Differential Equations 21 (1985), 11-18. Translation of Differentsial'nye Urav. 21 (1985), 15-23.
66. σ+1, Differential Equations 21 (1986), 751-758. Translation of Differentsial'nye Urav. 21 (1985), 1126-1134.
67. σ+1, Differential Equations 21 (1986), 751-758. Translation of Differentsial'nye Urav. 21 (1985), 1126-1134.
68. x + uβ near a "singular" point, Soviet Math. Dokl. 33 (1986), 840-844. Translation of Dokl. Akad. Nauk SSSR 288 (1986), 1293-1297.
69. x + uβ near a "singular" point, Soviet Math. Dokl. 33 (1986), 840-844. Translation of Dokl. Akad. Nauk SSSR 288 (1986), 1293-1297.
70. 64. V. A. GALAKTIONOV, Accurate estimates for the amplitude and support of unbounded solutions of the non-linear heat-conduction equation with a source, U.S.S.R. Comput. Math. Math. Phys. 30 (1990), 67-74. Translation of Zh. Vychisl. Mat. Mat. Fiz. 30 (1990), 438-448.
71. 64. V. A. GALAKTIONOV, Accurate estimates for the amplitude and support of unbounded solutions of the non-linear heat-conduction equation with a source, U.S.S.R. Comput. Math. Math. Phys. 30 (1990), 67-74. Translation of Zh. Vychisl. Mat. Mat. Fiz. 30 (1990), 438-448.
72. 65. V. A. GALAKTIONOV, V. A. DORODNITSYN, G. G. ELENIN, S. P. KURDYUMOV, AND A. A. SAMARSKII, A quasilinear heat equation with a source: Peaking, localization, symmetry exact solutions, asymptotics, structures, J. Soviet Math. 41 (1988), 1222-1292. Translation of Itog. Nauk. Tekhn. 28 (1986), 95-206.
73. 65. V. A. GALAKTIONOV, V. A. DORODNITSYN, G. G. ELENIN, S. P. KURDYUMOV, AND A. A. SAMARSKII, A quasilinear heat equation with a source: Peaking, localization, symmetry exact solutions, asymptotics, structures, J. Soviet Math. 41 (1988), 1222-1292. Translation of Itog. Nauk. Tekhn. 28 (1986), 95-206.
74. 66. V. A. GALAKTIONOV AND S. A. POSASHKOV, Monotonicity of a solution of a degenerate quasilinear parabolic equation, Differential Equations 26 (1991), 819-827. Translation of Differentsial'nye Urav. 26 (1990), 1127-1136.
75. 66. V. A. GALAKTIONOV AND S. A. POSASHKOV, Monotonicity of a solution of a degenerate quasilinear parabolic equation, Differential Equations 26 (1991), 819-827. Translation of Differentsial'nye Urav. 26 (1990), 1127-1136.
76. 67. C. S. GARDNER, J. M. GREENE, M. D. KRUSKAL, AND R. M. MIURA, Method for solving the Korteweg-deVries equation, Phys. Rev. Lett. 19 (1967), 1095-1097.
77. 68. B. H. GILDING, Properties of solutions of an equation in the theory of infiltration, Arch. Rational Mech. Anal. 65 (1977), 203-225.
78. 69. B. H. GILDING, A nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), 393-432.
79. 70. B. H. GILDING, Stabilization of flows through porous media, SIAM J. Math. Anal. 10 (1979), 237-246.
80. 71. B. H. GILDING, The occurrence of interfaces in nonlinear diffusion-advection processes, Arch. Rational Mech. Anal. 100 (1988), 243-263.
81. 72. B. H. GILDING, Improved theory for a nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), 165-224; 17 (1990), 479.
82. 72. B. H. GILDING, Improved theory for a nonlinear degenerate parabolic equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 16 (1989), 165-224; 17 (1990), 479.
83. 73. B. H. GILDING, Localization of solutions of a nonlinear Fokker-Planck equation with Dirichlet boundary conditions, Nonlinear Anal. 13 (1989), 1215-1240.
84. 74. B. H. GILDING, Qualitative mathematical analysis of the Richards equation, Transport Porous Med. 5 (1991), 651-666.
85. 75. B. H. GILDING, A singular nonlinear Volterra integral equation, J. Integral Equations Appl. 5 (1993), 465-502.
86. 76. B. H. GILDING AND R. KERSNER, Instantaneous shrinking in nonlinear diffusion-convection, Proc. Amer. Math. Soc. 109 (1990), 385-394.
87. 77. B. H. GILDING AND R. KERSNER, Diffusion-convection-réaction, frontières libres et une équation intégrale, C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 743-746.
88. 78. B. H. GILDING AND R. KERSNER, Travelling waves in nonlinear diffusion-advection-reaction, in preparation.
89. 79. R. E. GRUNDY, Asymptotic solution of a model non-linear convective diffusion equation, IMA J. Appl. Math. 31 (1983), 121-137.
90. 80. R. E. GRUNDY, The Cauchy problem for a nonlinear diffusion equation with absorption and convection, IMA J. Appl. Math. 40 (1988), 183-204.
91. 81. R. E. GRUNDY AND L. A. PELETIER, Short time behaviour of a singular solution to the heat equation with absorption, Proc. Roy. Soc. Edinburgh Sect. A 107 (1987), 271-288.
92. 82. M. GUEDDA, D. HILHORST, AND C. PICARD, The one-dimensional porous medium equation with convection: Continuous differentiability of interfaces after the waiting time, Appl. Math. Lett. 5 (1992), 59-62.
93. 83. W. S. C. GURNEY AND R. M. NISBET, The regulation of inhomogeneous populations, J. Theoret. Biol. 52 (1975), 441-457.
94. 84. M. E. GURTIN AND R. C. MACCAMY, On the diffusion of biological populations, Math. Biosci. 33 (1977), 35-49.
95. 85. M. A. HERRERO AND J. L. VÁZQUEZ, The one-dimensional nonlinear heat equation with absorption: Regularity of solutions and interfaces, SIAM J. Math. Anal. 18 (1987), 149-167.
96. 86. M. A. HERRERO AND J. L. VAZQUEZ, Thermal waves in absorbing media, J. Differential Equations 74 (1988), 218-233.
97. 87. K. HÖLLIG AND H.-O. KREISS, C∞-regularity for the porous medium equation, Math. Z. 192 (1986), 217-224.
98. 88. A. S. KALASHNIKOV, The occurrence of singularities in solutions of the non-steady see-page equation, U.S.S.R. Comput. Math. Math. Phys. 7 (1967), 269-275. Translation of Zh. Vychisl. Mat. Mat. Fiz. 7 (1967), 440-444.
99. 88. A. S. KALASHNIKOV, The occurrence of singularities in solutions of the non-steady see-page equation, U.S.S.R. Comput. Math. Math. Phys. 7 (1967), 269-275. Translation of Zh. Vychisl. Mat. Mat. Fiz. 7 (1967), 440-444.
100. 89. A. S. KALASHNIKOV, On equations of the nonstationary-filtration type in which the perturbation is propagated at infinite velocity, Moscow Univ. Math. Bull. 27 (1972), 104-108. Translation of Vestn. Moskov. Univ. Ser. I Mat. Mekh. 27 (1972), 45-49.
101. 89. A. S. KALASHNIKOV, On equations of the nonstationary-filtration type in which the perturbation is propagated at infinite velocity, Moscow Univ. Math. Bull. 27 (1972), 104-108. Translation of Vestn. Moskov. Univ. Ser. I Mat. Mekh. 27 (1972), 45-49.
102. 90. A. S. KALASHNIKOV, On the differential properties of generalized solutions of equations of the nonsteady-state filtration type, Moscow Univ. Math. Bull. 29 (1974), 48-53. Translation of Vestn. Moskov. Univ. Ser. I Mat. Mekh. 29 (1974), 62-68.
103. 90. A. S. KALASHNIKOV, On the differential properties of generalized solutions of equations of the nonsteady-state filtration type, Moscow Univ. Math. Bull. 29 (1974), 48-53. Translation of Vestn. Moskov. Univ. Ser. I Mat. Mekh. 29 (1974), 62-68.
104. 91. A. S. KALASHNIKOV, The propagation of disturbances in problems of non-linear heat conduction with absorption, U.S.S.R. Comput. Math. Math. Phys. 14 (1974), 70-85. Translation of Zh. Vychisl. Mat. Mat. Fiz. 14 (1974), 891-905.
105. 91. A. S. KALASHNIKOV, The propagation of disturbances in problems of non-linear heat conduction with absorption, U.S.S.R. Comput. Math. Math. Phys. 14 (1974), 70-85. Translation of Zh. Vychisl. Mat. Mat. Fiz. 14 (1974), 891-905.
106. 92. A. S. KALASHNIKOV, The nature of the propagation of perturbations in processes that can be described by quasilinear degenerate parabolic equations, Trud. Sem. Petrovsk. 1 (1975), 135-144. [In Russian]
107. 93. A. S. KALASHNIKOV, The effect of absorption on heat propagation in a medium in which the thermal conductivity depends on temperature, U.S.S.R. Comput. Math. Math. Phys. 16 (1976), 141-149. Translation of Zh. Vychisl. Mat. Mat. Fiz. 16 (1976), 689-696.
108. 93. A. S. KALASHNIKOV, The effect of absorption on heat propagation in a medium in which the thermal conductivity depends on temperature, U.S.S.R. Comput. Math. Math. Phys. 16 (1976), 141-149. Translation of Zh. Vychisl. Mat. Mat. Fiz. 16 (1976), 689-696.
109. 94. A. S. KALASHNIKOV, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russ. Math. Surveys 42 (1987), 169-222. Translation of Uspekhi Mat. Nauk 42 (1987), 135-176.
110. 94. A. S. KALASHNIKOV, Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations, Russ. Math. Surveys 42 (1987), 169-222. Translation of Uspekhi Mat. Nauk 42 (1987), 135-176.
111. 95. S. KAMENOMOSTSKAYA, The asumptotic behaviour of the solution of the filtration equation, Israel J. Math. 14 (1973), 76-87.
112. 96. S. KAMIN (KAMENOMOSTSKAYA), Similar solutions and the asymptotics of filtration equations, Arch. Rational Mech. Anal. 60 (1976), 171-183.
113. 97. R. KERSHNER, Localization conditions for thermal perturbations in a semibounded moving medium with absorption, Moscow Univ. Math. Bull 31 (1976), 90-95. Translation of Vestn. Moskov. Univ. Ser. I Mat. Mekh. 31 (1976), 52-58.
114. 97. R. KERSHNER, Localization conditions for thermal perturbations in a semibounded moving medium with absorption, Moscow Univ. Math. Bull 31 (1976), 90-95. Translation of Vestn. Moskov. Univ. Ser. I Mat. Mekh. 31 (1976), 52-58.
115. 98. R. KERSHNER, Some properties of generalized solutions of quasilinear parabolic equations, Acta Math. Acad. Sci. Hungar. 32 (1978), 301-330. [In Russian]
116. 99. R. KERSHNER, The behavior of the solutions of degenerate quasilinear parabolic equation as t → ∞, Acta Math. Acad. Sci. Hungar. 34 (1979), 157-163. [In Russian]
117. 100. R. KERSNER, The behavior of temperature fronts in media with nonlinear thermal conductivity under absorption, Moscow Univ. Math. Bull. 33 (1978), 35-41. Translation of Vestn. Moskov. Univ. Ser. I Mat. Mekh. 33 (1978), 44-51.
118. 100. R. KERSNER, The behavior of temperature fronts in media with nonlinear thermal conductivity under absorption, Moscow Univ. Math. Bull. 33 (1978), 35-41. Translation of Vestn. Moskov. Univ. Ser. I Mat. Mekh. 33 (1978), 44-51.
119. 101. R. KERSNER, Degenerate parabolic equations with general nonlinearities, Nonlinear Anal. 4 (1980), 1043-1062.
120. 102. R. KERSNER, Filtration with absorption: Necessary and sufficient condition for the propagation of perturbations to have finite velocity, J. Math. Anal. Appl. 90 (1982), 463-479.
121. 103. R. KERSNER, Nonlinear heat conduction with absorption: Space localization and extinction in finite time, SIAM J. Appl. Math. 43 (1983), 1274-1285.
122. 104. N. V. KHUSNYTDINOVA, The limiting moisture profile during infiltration in to a homogeneous soil, J. Appl. Math. Mech. 31 (1967), 783-789. Translation of Prikl. Mat. Mekh. 31 (1967), 770-776.
123. 104. N. V. KHUSNYTDINOVA, The limiting moisture profile during infiltration in to a homogeneous soil, J. Appl. Math. Mech. 31 (1967), 783-789. Translation of Prikl. Mat. Mekh. 31 (1967), 770-776.
124. 105. A. KOLMOGOROFF, I. PETROVSKY, AND N. PISCOUNOFF, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application a un problème biologique, Bull. Univ. Moscou Sér. Internat. Sect. A Math. Méc. 1 (1937), 1-25.
125. 106. D. J. KORTEWEG AND G. DE VRIES, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39 (1895), 422-443.
126. 107. O. A. LADYZHENSKAJA, V. A. SOLONNIKOV, AND N. N. URAL'CEVA, "Linear and Quasi-Linear Equations of Parabolic Type," American Mathematical Society, Providence, RI, 1968.
127. 108. L. K. MARTINSON AND K. B. PAVLOV, The problem of the three-dimensional localization of thermal perturbations in the theory of non-linear heat conduction, U.S.S.R. Comput. Math. Math. Phys 12 (1972), 261-268. Translation of Zh. Vychisl. Mat. Mat. Fiz. 12 (1972), 1048-1053.
128. 108. L. K. MARTINSON AND K. B. PAVLOV, The problem of the three-dimensional localization of thermal perturbations in the theory of non-linear heat conduction, U.S.S.R. Comput. Math. Math. Phys 12 (1972), 261-268. Translation of Zh. Vychisl. Mat. Mat. Fiz. 12 (1972), 1048-1053.
129. 109. V P. MASLOV, V. G. DANILOV, AND K. A. VOLOSOV, "Mathematical Modeling of Heat-and Mass-Transport Processes," Nauka, Moscow, 1987. [In Russian]
130. 110. D. MOLLISON, Spatial contact models for ecological and epidemic spread, J. Roy. Statist. Soc. Ser. B 39 (1977), 283-326.
131. 111. J. D. MURRAY, "Lectures on Nonlinear-Differential-Equation Models in Biology," Clarendon Press, Oxford, 1977.
132. 112. J. NAGUMO, S. YOSHIZAWA, AND S. ARIMOTO, Bistable transmission lines, IEEE Trans. Circuit Theory 12 (1965), 400-412.
133. 113. G. NICOLIS AND I. PRIGOGINE, "Self-Organization in Nonequilibrium Systems," Wiley, New York, 1977.
134. 114. A. OKUBO, "Diffusion and Ecological Problems: Mathematical Models," Springer-Verlag, Berlin, 1980.
135. 115. O. A. OLEINIK, A. S. KALASHNIKOV, AND CHZHOU Y.-L., The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 667-704. [In Russian]
136. 116. O. A. OLEINIK AND S. N. KRUZHKOV, Quasi-linear second-order parabolic equations with many independent variables, Russ. Math. Surv. 16 (1961), 105-146.
137. Translation of Uspekhi Mat. Nauk 16 (1961), 115-155.
138. 117. A. DE PABLO AND J. L. VAZQUEZ, The balance between strong reaction and slow diffusion, Comm. Partial Differential Equations 15 (1990), 159-183.
139. 118. A. DE PABLO AND J. L. VAZQUEZ, Travelling waves and finite propagation in a reaction-diffusion equation, J. Differential Equations 93 (1991), 19-61.
140. 119. A. DE PABLO AND J. L. VAZQUEZ, An overdetermined initial and boundary-value problem for a reaction-diffusion equation, Nonlinear Anal. 19 (1992), 159-269.
141. 120. R. E. PATTLE, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407-409.
142. 121. L. A. PELETIER, A necessary and sufficient condition for the existence of an interface in flows through porous media, Arch. Ration. Mech. Anal. 56 (1974), 183-190.
143. 122. L. A. PELETIER, On the existence of an interface in nonlinear diffusion processes, in "Ordinary and Partial Differential Equations" (B. D. Sleeman and I. M. Michael, Eds.), pp.412-416, Lecture Notes in Mathematics, Vol.415, Springer-Verlag, Berlin, 1974.
144. 123. L. A. PELETIER, The porous media equation, in "Applications of Nonlinear Analysis in the Physical Sciences" (H. Amann, N. Bazley, and K. Kirchgässner, Eds.), pp. 229-241, Pitman, Boston, 1981.
145. 124. B. PERTHAME AND J. L. VAZQUEZ, Bounded speed of propagation for solutions to radiative transfer equations, Comm. Math. Phys. 130 (1990), 457-469.
146. 125. D. PHILLIPS, Existence of solutions of quenching problems, Appl. Anal. 24 (1987), 253-264.
147. 126. S. D. POISSON, "Théorie Mathématique de la Chaleur," Bachelier, Paris, 1835.
148. 127. L. D. POKROVSKII AND S. N. TARANENKO, Conditions for space localization of the solutions of the non-linear equation of heat conduction, U.S.S.R. Comput. Math. Math. Phys. 22 (1982), 264-269. Translation of Zh. Vychisl. Mat. Mat. Fiz. 22 (1982), 747-751.
149. 127. L. D. POKROVSKII AND S. N. TARANENKO, Conditions for space localization of the solutions of the non-linear equation of heat conduction, U.S.S.R. Comput. Math. Math. Phys. 22 (1982), 264-269. Translation of Zh. Vychisl. Mat. Mat. Fiz. 22 (1982), 747-751.
150. 128. L. A. RICHARDS, Capillary conduction of liquids through porous mediums, Physics 1 (1931), 318-333.
151. 129. P. ROSENAU AND S. KAMIN, Thermal waves in an absorbing and convecting medium, Phys. D 8 (1983), 273-283.
152. 130. P. E. SACKS, The initial and boundary value problem for a class of degenerate parabolic equations, Comm. Partial Differential Equations 8 (1983), 693-733.
153. 131. A. SCHATZ, Free boundary problems of Stephan type with prescribed flux, J. Math. Anal. Appl. 28 (1969), 569-580.
154. 132. A. C. SCOTT, The electrophysics of a nerve fiber, Rev. Modern Phys. 47 (1975), 487-533.
155. 133. S. I. SHMAREV, Instantaneous appearance of singularities of a solution of a degenerate parabolic equation, Siberian Math. J. 31 (1991), 671-682. Translation of Sibirsk. Mat. Zh. 31 (1990), 166-179.
156. 133. S. I. SHMAREV, Instantaneous appearance of singularities of a solution of a degenerate parabolic equation, Siberian Math. J. 31 (1991), 671-682. Translation of Sibirsk. Mat. Zh. 31 (1990), 166-179.
157. 1, J. Math. Soc. Japan 36 (1984), 177-189.
158. 135. J. SMOLLER, "Shock Waves and Reaction-Diffusion Equations," Springer-Verlag, New York, 1983.
159. 136. SONG B., Existence, uniqueness and properties of the solutions of a degenerate parabolic equation with diffusion-advection-absorption, Acta Math. Appl. Sinica (English Ser.) 10 (1994), 113-140.
160. 137. D. SWARTZENDRUBER, The flow of water in unsaturated soils, in "Flow through Porous Media" (R. J. M. de Wiest, Ed.), pp. 215-292, Academic Press, New York, 1969.
161. 138. J. L. VAZQUEZ, Asymptotic behaviour and propagation properties on the one-dimensional flow of gas in a porous medium, Trans. Amer. Math. Soc. 277 (1983), 507-527.
162. 139. J. L. VAZQUEZ, The interfaces of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 285 (1984), 717-737.
163. 140. J. L. VAZQUEZ, Behaviour of the velocity of one-dimensional flows in porous media, Trans. Amer. Math. Soc. 286 (1984), 787-802.
164. 141. J. L. VAZQUEZ, Regularity of solutions and interfaces of the porous medium equation via local estimates, Proc. Roy. Soc. Edinburgh Sect. A 112 (1989), 1-13.
165. 142. J. L. VAZQUEZ, An introduction to the mathematical theory of the porous medium equation, in Shape Optimization and Free Boundaries(M. C. Delfour and G. Sabidussi, Eds.), pp. 347-389, Kluwer Academic, Dordrecht, 1992.
166. 143. IA. B. ZEL'DOVICH AND G. I. BARENBLATT, The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations, Sov. Phys. Dokl. 3 (1958), 44-47. Translation of Dokl. Akad. Nauk SSSR (N.S.) 118 (1958), 671-674.
167. 143. IA. B. ZEL'DOVICH AND G. I. BARENBLATT, The asymptotic properties of self-modelling solutions of the nonstationary gas filtration equations, Sov. Phys. Dokl. 3 (1958), 44-47. Translation of Dokl. Akad. Nauk SSSR (N.S.) 118 (1958), 671-674.
168. 144. Y. B. ZELDOVICH, "Theory of Flame Propagation," National Advisory Committee for Aeronautics Technical Memorandum 1282, 1951. Translation of Zh. Fiz. Khim. 22 (1948), 27-49.
169. 144. Y. B. ZELDOVICH, "Theory of Flame Propagation," National Advisory Committee for Aeronautics Technical Memorandum 1282, 1951. Translation of Zh. Fiz. Khim. 22 (1948), 27-49.
170. 145. YA. B. ZEL'DOVICH AND A. S. KOMPANEEC, On the theory of propagation of heat with the heat conductivity depending upon the temperature, in "Collection in Honour of the Seventieth Birthday of Academician A. F. Ioffe," pp. 61-71, Iz. Akad. Nauk SSSR Moscow, 1950. [In Russian]
171. 146. YA. B. ZEL'DOVICH AND YU. P. RAIZER, "Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena Volume I," Academic Press, New York, 1966.
172. 147. YA. B. ZEL'DOVICH AND YU. P. RAIZER, "Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena Volume II," Academic Press, New York, 1967.
173. 148. J. ZHAO, Uniqueness of solutions of quasilinear degenerate parabolic equations, Dongbei Shuxue 1 (1985), 153-165.