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A necessary and sufficient condition for finite speed of propagation in the theory of doubly nonlinear degenerate parabolic equations

Published online by Cambridge University Press:  14 November 2011

B. H. Gilding  and
R. Kersner
B. H. Gilding
Affiliation:
Faculty of Applied Mathematics, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
R. Kersner
Affiliation:
Computer and Automation Research Institute, Hungarian Academy of Sciences, P.O. Box 63, H-1518 Budapest, Hungary
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Abstract

A degenerate parabolic partial differential equation with a time derivative and first- and second-order derivatives with respect to one spatial variable is studied. The coefficients in the equation depend nonlinearly on both the unknown and the first spatial derivative of a function of the unknown. The equation is said to display finite speed of propagation if a non-negative weak solution which has bounded support with respect to the spatial variable at some initial time, also possesses this property at later times. A criterion on the coefficients in the equation which is both necessary and sufficient for the occurrence of this phenomenon is established. According to whether or not the criterion holds, weak travelling-wave solutions or weak travelling-wave strict subsolutions of the equation are constructed and used to prove the main theorem via a comparison principle. Applications to special cases are provided.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 126 , Issue 4 , 1996 , pp. 739 - 767
Copyright
Copyright © Royal Society of Edinburgh 1996

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References

1Alt, H. W. and Luckhaus, S.. Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311–41.Google Scholar
2Antoncev, S. N.. The character of perturbations described by solutions of multidimensional degenerate parabolic equations. Dinamika Sploshn. Sredy 40 (1979), 114–22 (in Russian).Google Scholar
3Antoncev, S. N.. On the localization of solutions of nonlinear degenerate elliptic and parabolic equations. Soviet Math. Dokl. 24 (1981), 420–24; originally appearing in Russian as: Dokl. Akad. Nauk SSSR 260 (1981), 1289–93.Google Scholar
4Antontsev, S. N.. Metastable localization of solutions of degenerate parabolic equations of general form. Dinamika Sploshn. Sredy 83 (1987), 138–44 (in Russian).Google Scholar
5Antontsev, S. N. and Dias, Kh. I.. New results on localization of solutions of nonlinear elliptic and parabolic equations obtained by the energy method. Soviet Math. Dokl. 38 (1989), 535–9; originally appearing in Russian as: Dokl. Akad. Nauk SSSR 303 (1988), 524–9.Google Scholar
6Antontsev, S. N. and Shmarëv, S. I.. The local energy method and vanishing of weak solutions of nonlinear parabolic equations. Soviet Math. Dokl. 43 (1991), 738–42; originally appearing in Russian as: Dokl. Akad. Nauk SSSR 318 (1991), 777–81.Google Scholar
7Atkinson, C. and Bouillet, J. E.. Some qualitative properties of solutions of a generalised diffusion equation. Math. Proc. Cambridge Philos. Soc. 86 (1979), 495–510.CrossRefGoogle Scholar
8Bamberger, A.. Étude d'une équation doublement non linéaire. J. Fund. Anal. 24 (1977), 148–55.CrossRefGoogle Scholar
9Barenblatt, G. I.. On self-similar motions of a compressible fluid in a porous medium. Prikl. Mat. Mekh. 16 (1952), 679–98 (in Russian).Google Scholar
10Bertsch, M. and Passo, R. Dal. Hyperbolic phenomena in a strongly degenerate parabolic equation. Arch. Rational Mech. Anal. 117 (1992), 349–87.CrossRefGoogle Scholar
11Bertsch, M. and Passo, R. Dal. A parabolic equation with a mean-curvature type operator. In Nonlinear Diffusion Equations and Their Equilibrium States 3, eds Lloyd, N. G., Ni, W. M., Peletier, L. A. and Serrin, J., 89–97 (Boston: Birkhäuser, 1992).CrossRefGoogle Scholar
12Bertsch, M., van, C. J. Duijn, Esteban, J. R. and Zhang, H.. Regularity of the free boundary in a doubly degenerate parabolic equation. Comm. Partial Differential Equations 14 (1989), 391–412.CrossRefGoogle Scholar
13Bertsch, M., Esteban, J. R. and Zhang, H.. On the asymptotic behaviour of the solutions of a degenerate equation in hydrology. Nonlinear Anal. 19 (1992), 365–74.CrossRefGoogle Scholar
14Blanc, P.. A degenerate parabolic equation. In Semigroup Theory and Applications, eds Clément, P., Invernizzi, S., Mitidieri, E. and Vrabie, I. I., 59–65 (New York: Marcel Dekker, 1989).Google Scholar
15Blanc, P.. Existence de solutions discontinues pour des équations paraboliques. C. R. Acad. Sci. Paris Sér. I Math. 310 (1990), 53–6.Google Scholar
16Blanc, P.. On the regularity of the solutions of some degenerate parabolic equations. Comm. Partial Differential Equations 18 (1993), 821–46.CrossRefGoogle Scholar
17Bouillet, J. E. and Atkinson, C.. A generalized diffusion equation: radial symmetries and comparison theorems. J. Math. Anal. Appl. 95 (1983), 37–68.CrossRefGoogle Scholar
18Passo, R. Dal. Uniqueness of the entropy solution of a strongly degenerate parabolic equation. Comm. Partial Differential Equations 18 (1993), 265–79.CrossRefGoogle Scholar
19Diaz, J. I. and Thelin, F. de. On a nonlinear parabolic problem arising in some models related to turbulent flows. SIAM J. Math. Anal. 25 (1994), 1085–111.CrossRefGoogle Scholar
20Diaz, J. I. and Veron, L.. Local vanishing properties of solutions of elliptic and parabolic quasilinear equations. Trans. Amer. Math. Soc. 290 (1985), 787–814.CrossRefGoogle Scholar
21DiBenedetto, E.. Degenerate Parabolic Equations (New York: Springer, 1993).CrossRefGoogle Scholar
22Duijn, C. J. van and Zhang, H.. Regularity properties of a doubly degenerate equation in hydrology. Comm. Partial Differential Equations 13 (1988), 261–319.CrossRefGoogle Scholar
23Duyn, C. J. van. Some fundamental properties of the simultaneous flow of fresh and salt groundwater in horizontally extended aquifers. In Flow and Transport in Porous Media, eds Verruijt, A. and Barends, F. B. J., 83–90 (Rotterdam: A. A. Balkema, 1981).Google Scholar
24Duyn, C. J. van and Hilhorst, D.. On a doubly nonlinear diffusion equation in hydrology. Nonlinear Anal. 11 (1987), 305–33.CrossRefGoogle Scholar
25Esteban, J. R. and Vazquez, J. L.. On the equation of turbulent filtration in one-dimensional porous media. Nonlinear Anal. 10 (1986), 1303–25.CrossRefGoogle Scholar
26Esteban, J. R. and Vazquez, J. L.. Homogeneous diffusion in R with power-like nonlinear diffusivity. Arch. Rational Mech. Anal. 103 (1988), 39–80.CrossRefGoogle Scholar
27Gilding, B. H.. The occurrence of interfaces in nonlinear diffusion-advection processes. Arch. Rational Mech. Anal. 100 (1988), 243–63.CrossRefGoogle Scholar
28Gilding, B. H.. A singular nonlinear Volterra integral equation. J. Integral Equations Appl. 5 (1993), 465–502.CrossRefGoogle Scholar
29Gilding, B. H. and Kersner, R.. Diffusion-convection-réaction, frontiéres libres et une équation integrate. C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), 743–46.Google Scholar
30Gilding, B. H. and Kersner, R.. The characterization of reaction-convection-diffusion processes by travelling waves. J. Differential Equations, 124 (1996), 27–79.CrossRefGoogle Scholar
31Ivanov, A. V.. Quasilinear Degenerate and Nonuniformly Elliptic and Parabolic Equations of Second Order (Providence, Rhode Island: American Mathematical Society, 1984); originally appearing in Russian as: Trudy Mat. Inst. Steklov. 160 (1982), 5–285.Google Scholar
32Jong, G. de Josselin de. The simultaneous flow of fresh and salt water in aquifers of large horizontal extension determined by shear flow and vortex theory. In Flow and Transport in Porous Media, eds Verruijt, A. and Barends, F. B. J., 75–82 (Rotterdam: A. A. Balkema, 1981).Google Scholar
33Kačur, J.. On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolev spaces I. Math. Z. 203(1990), 153–71.CrossRefGoogle Scholar
34Kačur, J.. On a solution of degenerate elliptic-parabolic systems in Orlicz-Sobolev spaces II. Math. Z. 203(1990), 569–79.CrossRefGoogle Scholar
35Kalashnikov, A. D.. The concept of a finite rate of propagation of a perturbation. Russian Math. Surveys 34 (1979), 235–36; originally appearing in Russian as: Uspekhi Mat. Nauk 34 (1979), 199–200.CrossRefGoogle Scholar
36Kalashnikov, A. S.. A nonlinear equation arising in the theory of nonlinear filtration. Trudy Sem. Petrovsk. 4 (1978), 137–46 (in Russian).Google Scholar
37Kalashnikov, A. S.. On the propagation of perturbations in the processes described by degenerate parabolic equations with non-power nonlinearities. Trudy Sem. Petrovsk. 10 (1984), 118–34, 238 (in Russian).Google Scholar
38Kalashnikov, A. S.. Propagation of perturbations in the first boundary-value problem for a doublenonlinear degenerate parabolic equation. J. Soviet Math. 32 (1986), 315–20; originally appearing in Russian as: Trudy Sem. Petrovsk. 8 (1982), 128–34.CrossRefGoogle Scholar
39Kalashnikov, A. S.. Cauchy problem for second-order degenerate parabolic equations with nonpower nonlinearities. J. Soviet Math. 33 (1986), 1014–25; originally appearing in Russian as: Trudy Sem. Petrovsk. 6(1981), 83–96.CrossRefGoogle Scholar
40Kalashnikov, A. S.. Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations. Russian Math. Surveys 42 (1987), 169–222; originally appearing in Russian as: Uspekhi Mat. Nauk 42 (1987), 135–76.CrossRefGoogle Scholar
41Leibenson, L. S.. Turbulent movement of gas in a porous medium. Izv. Akad. Nauk SSSR Ser. Geograf. Geofiz. 9 (1945), 3–6 (in Russian).Google Scholar
42Leibenson, L. S.. General problem of the movement of a compressible fluid in a porous medium. Izv. Akad. Nauk SSSR Ser. Geograf. Geofiz. 9 (1945), 7–10 (in Russian).Google Scholar
43Leibenson, L. S.. A fundamental law of gas motion through a porous medium. C. R. Acad. Sci. URSS 47 (1945), 16–18.Google Scholar
44Leibenzon, L. S.. The Motion of Natural Liquids and Gases in Porous Media (Moscow: OGIZ Gosudarstvenoe IzdatePstva Tekhniko-Teoreticheskoi Literatury, 1947 (in Russian)).Google Scholar
45Lions, J. L.. Quelques Méthodes de Résolution des Problémes aux Limites Non Linéaires (Paris: Dunod, 1969).Google Scholar
46Marinov, M. L. and Rangelov, T. V.. Estimates on the support of the solutions for some nonlinear degenerate parabolic equations. C. R. Acad. Bulgare Sci. 39 (1986), 17–19.Google Scholar
47Nagai, T.. On solutions with compact support of certain nonlinear parabolic equations. Boll. Un. Mat. Ital.B 15 (1978), 9–19.Google Scholar
48Pavlov, K. B. (ed.). Questions of the Theory of Nonlinear Transport Processes, Trudy MVTU 374 (Moscow: Moskovskoe Vysshee Tekhnicheskoe Uchilishe, 1981 (in Russian)).Google Scholar
49Pavlov, K. B.. Pokrovskii, L. D. and Taranenko, S. N.. Properties of solutions of a nonlinear transfer equation. Differential Equations 17 (1982), 1063–8; originally appearing in Russian as: Differenstial'nye Uravneniya 17 (1981), 1661–7.Google Scholar
50Pokrovskii, L. D. and Taranenko, S. N.. Conditions for space localization of the solutions of the nonlinear equation of heat conduction. U.S.S.R. Comput. Math, and Math. Phys. 22 (1983), 264–9; originally appearing in Russian as: Zh. Vychisl. Mat. i Mat. Fiz. 22 (1982), 747–51.CrossRefGoogle Scholar
51Porzio, M. M. and Vespri, V.. Holder estimates for local solutions of some doubly nonlinear degenerate parabolic equations. J. Differential Equations 103 (1993), 146–78.CrossRefGoogle Scholar
52Raviart, P. A.. Sur la résolution et l'approximation de certaines équations paraboliques non linéaires dégénérées. Arch. Rational Mech. Anal. 25 (1967), 64–80.CrossRefGoogle Scholar
53Rykov, Yu. G.. The finite propagation rate of perturbations and momentum for the generalized solution of quasilinear parabolic equations. In Differential Equations and their Applications, 101–5 (Moscow: Moskovskogo Universiteta, 1984, (in Russian)).Google Scholar
54YRykov, u. G.. The behaviour with increasing time of the generalized solutions of degenerate parabolic equations with nonisotropic nonlinearities not of power type. In Differential Equations and their Applications, 160–4 (Moscow: Moskovskogo Universiteta, 1984 (in Russian)).Google Scholar
55Su, N.. Propagation properties for a nonlinear degenerate diffusion equation. In International Symposium on Nonlinear Problems in Engineering and Science, eds Xiao, S. and Hu, X.-C., 137–47 (Beijing: Science Press, 1992).Google Scholar
56Su, N.. On propagation properties of perturbation in nonlinear degenerate diffusions with convection and absorption. Nonlinear Anal. 23 (1994), 1321–30.Google Scholar
57Tsutsumi, M.. On solutions of some doubly nonlinear degenerate parabolic equations with absorption. J. Math. Anal. Appl. 132 (1988), 187–212.CrossRefGoogle Scholar
58Vespri, V.. Analytic semigroups, degenerate elliptic operators and applications to nonlinear Cauchy problems. Ann. Mat. Pura Appl. (4) 155 (1989), 353–88.CrossRefGoogle Scholar
59Vespri, V.. Abstract quasilinear parabolic equations with variable domains. Differential Integral Equations 4 (1991), 1041–72.CrossRefGoogle Scholar
60Vespri, V.. On the local behaviour of solutions of certain singular parabolic equations. Nonlinear Anal. 16(1991), 827–46.CrossRefGoogle Scholar
61Vespri, V.. On the local behaviour of solutions of a certain class of doubly nonlinear parabolic equations. Manuscripta Math. 75 (1992), 65–80.CrossRefGoogle Scholar
62Yin, J.. On a class of quasilinear parabolic equations of second order with double-degeneracy. J. Partial Differential Equations 4 (1990), 49–64.Google Scholar
63Yin, J.. Solutions with compact support for nonlinear diffusion equations. Nonlinear Anal. 19 (1992), 309–21.Google Scholar
64Yin, J.. Parabolic equations with both non-convexity nonlinearities and strong degeneracy. Ada Math. Sci. (English Edn.), 14 (1994), 451–60.Google Scholar