# On some nonlinear problems in mathematical physics with exponents close to critical ones

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## Abstract

The paper deals with several problems having the following common feature. The differential equation under consideration contains a power function of the solution or of its gradient, with the exponent depending on a small parameter. This dependence is such that a certain property of the solutions corresponding to the positive values of the parameter may disappear as the latter tends to zero. This phenomenon is shown to be lacking in the case where not only the said exponent, but also the coefficients of the equation depend on the small parameter in a suitable way. Several theorems are proved on sharp conditions, ensuring such uniformity of various properties exhibited by the solutions. Bibliography: 44 titles.

### Keywords

Differential Equation Mathematical Physic Power Function Small Parameter Nonlinear Problem## Preview

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### References

- 1.L. S. Leibenzon,
*Motion of Natural Fluids and Gases in a Porous Medium*[in Russian], Gostekhizdat, Moscow (1947).Google Scholar - 2.H. Schlichting,
*Boundary Layer Theory*, McGraw-Hill, New York-Toronto-London (1968).Google Scholar - 3.M. E. Gurtin and R. C. MacCamy, “On the diffusion of biological populations,”
*Math. Biosciences*,**33**, No. 1, 35–49. (1977).MathSciNetGoogle Scholar - 4.Ya. B. Zel'dovich and A. S. Kompaneets, “On the theory of heat propagation with the heat conductivity depending on the temperature,” in:
*Collection in Honour of the 70th Birthday of Academician A. F. Ioffe*, Izd. Akad. Nauk SSSR, Moscow (1950), pp. 61–71.Google Scholar - 5.H. Breźis, “On some nonlinear degenerate parabolic equations,” in:
*Nonlinear Functional Analysis*(Proc. Symp. Pure Math. Chicago, 1968,**18**, Part 1), Amer. Math. Soc., Providence (1970), pp. 28–38.Google Scholar - 6.O. A. Oleinik, “On the equations of unsteady filtration type,”
*Dokl. Akad. Nauk SSSR*,**113**, No. 6, 1210–1213 (1957).MATHMathSciNetGoogle Scholar - 7.A. S. Kalashnikov, “Some problems of the qualitative theory of nonlinear second-order degenerate parabolic equations,”
*Usp. Mat. Nauk*,**42**, No. 2, 135–176 (1987).MATHMathSciNetGoogle Scholar - 8.J.-L. Lions,
*Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires*Dunod, Gauthier-Villars, Paris (1969).Google Scholar - 9.J. I. Diaz,
*Nonlinear Partial Differential Equations and Free Boundaries. Volume 1*(*Research Notes in Mathematics*,**106**), Pitman, London (1985).Google Scholar - 10.N. V. Krylov,
*Non-linear Second-Order Elliptic and Parabolic Equations*[in Russian], Nauka, Moscow (1985).Google Scholar - 11.S. N. Antontsev,
*Localization for Solutions of Degenerate Equations in Mechanics of Continua*[in Russian], Institute of Hydrodynamics, Novosibirsk (1986).Google Scholar - 12.A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov,
*Blow-up in Quasi-linear Parabolic Equations*[in Russian], Nauka, Moscow (1987). English translation: Walter de Gruyter, Berlin (1994).Google Scholar - 13.A. Friedman,
*Variational Principles and Free-Boundary Problems*, John Wiley and Sons, New York (1982).Google Scholar - 14.M. Bertcsh and L. A. Peletier,
*Porous Media Equations: an Overview*, Math. Inst. Leiden, Univ. of Leiden, Report No. 7 (1983).Google Scholar - 15.D. G. Aronson, “The porous medium equation,”
*Lect. Notes Math.***1224**, 1–46 (1986).MATHMathSciNetGoogle Scholar - 16.O. A. Oleinik, M. Primicerio, and E. V. Radkevich, “Stefan-like problems,”
*Meccanica*,**28**, 129–143 (1993).CrossRefGoogle Scholar - 17.A. V. Ivanov, “Quasi-linear parabolic equations with possible double degeneracy,”
*Algebra Analiz*,**4**, No. 6 114–130 (1992).MATHGoogle Scholar - 18.J. I. Diaz and L. Véron, “Local vanishing properties of solutions of elliptic and parabolic quasi-linear equations,”
*Trans. Amer. Math. Soc.*,**20**, No. 2, 787–814 (1985).Google Scholar - 19.E. S. Sabinina, “On a class of nonlinear degenerate parabolic equations,”
*Dokl. Akad. Nauk SSSR*,**143**, No. 4, 794–797 (1962).MATHMathSciNetGoogle Scholar - 20.S. N. Antontsev and J. I. Diaz, “New results on the localization for solutions of nonlinear elliptic and parabolic equations via the energy method,”
*Dokl. Akad. Nauk SSSR*,**303**, No. 3, 524–529 (1988).Google Scholar - 21.M. Tsutsumi, “On solutions of some doubly nonlinear degenerate parabolic equations with absorption,”
*J. Math. Anal. Appl.*,**132**, No. 1, 187–212 (1988).MATHMathSciNetGoogle Scholar - 22.Y. C. Kwong, “Asymptotic behavior of a plasma type equation with finite extinction,”
*Arch. Rat. Mech. Anal.*,**104**, No. 3, 277–294 (1988).MATHMathSciNetGoogle Scholar - 23.Song Binghehg,
*The Existence, Uniqueness and Properties of Solutions of a Degenerate Parabolic Equation with Diffusion-Advection-Absorption*, Science Report No. 88005, Department of Applied Math., Tsinghua University, Beijing (1988).Google Scholar - 24.A. Friedman and M. A. Herrero, “Extinction and positivity for a system of semilinear parabolic variational inequalities,”
*J. Math. Anal. Appl.*,**167**, No. 1, 167–175 (1992).MathSciNetGoogle Scholar - 25.S. N. Antontsev and S. I. Shmarev, “The local energy method and vanishing of weak solutions of nonlinear parabolic equations,”
*Dokl. Akad. Nauk SSSR*,**318**, No. 4, 777–780 (1991).MathSciNetGoogle Scholar - 26.V. V. Chistyakov, “On properties of solutions of semilinear second-order parabolic equations,”
*Trudy Seminara imeni I. G. Petrovskogo*, No. 15, 70–107 (1991).MATHMathSciNetGoogle Scholar - 27.R. Kersner and F. Nicolosi, “The nonlinear heat conduction with absorption: effects of variable coefficients,”
*J. Math. Anal. Appl.*,**170**, No. 2, 551–566 (1992).MathSciNetGoogle Scholar - 28.J. J. L. Velazquez, V. A. Galaktionov S. A. Posashkov, and M. A. Herrero, “On a general approach to extinction and blow-up for quasi-linear heat equations,”
*Zh. Vychisl. Mat. Mat. Fiz.*,**33**, No. 2, 246–258 (1993).MathSciNetGoogle Scholar - 29.V. A. Galaktionov and J. L. Velazquez, “Extinction for a quasi-linear heat equation with absorption. I. Technique of intersection comparison,”
*Commun. Part. Diff. Eq.*,**19**, No.7&8, 1075–1106 (1994).Google Scholar - 30.A. S. Kalashnikov, “Nonlinear phenomena in nonstationary processes described by asymptotically linear equations,”
*Differents. Uravn.*,**29**, No. 3, 381–391 (1993).MATHMathSciNetGoogle Scholar - 31.Yu. G. Rykov, “On the propagation of perturbations for essentially nonautonomous quasi-linear first-order equations,”
*Trudy Seminara imeni I. G. Petrovskogo*, No. 17, 89–117 (1994).MATHMathSciNetGoogle Scholar - 32.L. K. Martinson and K. B. Pavlov, “On the problem of spatial localization of thermal perturbations in the theory of nonlinear heat conduction,”
*Zh. Vychisl. Mat. Mat. Fiz.*,**12**, No. 4 1048–1053 (1972).MathSciNetGoogle Scholar - 33.M. Mimura and T. Nagai,
*Asymptotic behavior of the interface to a certain nonlinear diffusion-advection, equation, Nonlinear Parabolic Equations: Qualitative Properties of Solutions*(Research Notes in Mathematics,**149**), Longman, Harlow (1987), pp. 156–161.Google Scholar - 34.B. H. Gilding, “Localization of solutions of a nonlinear Fokker-Planck equation with Dirichlet boundary conditions,”
*Nonlinear Analysis*,**13**, No. 10, 1215–1240 (1989).MATHMathSciNetGoogle Scholar - 35.G. Gagneux and M. Madaune-Tort, “Three-dimensional solutions of nonlinear degenerate diffusion-convection processes,”
*Eur. J. Appl. Math.*,**2**, No. 2, 171–187 (1991).MathSciNetGoogle Scholar - 36.R. Natalini and A. Tesei “On a class of perturbed conservation laws,”
*Adv. Appl. Math.*,**13**, 429–453 (1992).CrossRefMathSciNetGoogle Scholar - 37.U. G. Abdullaev, “Localization of unbounded solutions of a nonlinear heat equation with convection,”
*Dokl. Akad. Nauk*,**329**, No. 3, 535–537 (1993).MATHMathSciNetGoogle Scholar - 38.E. M. Landis, “On the ‘dead core’ for semilinear degenerate elliptic inequalities,”
*Differents. Uravn.*,**29**No. 3, 414–423 (1993).MATHMathSciNetGoogle Scholar - 39.S. G. Krein and M. I. Khazan, “Differential equations in Banach space,”
*Itogi Nauki i Tekhniki. Mat. Analiz*,**21**130–264 (1983).MathSciNetGoogle Scholar - 40.A. Friedman and K. Höllig, “On the mesa problem,”
*J. Math. Anal. Appl.*,**132**, No. 1 564–571 (1987).Google Scholar - 41.Ph. Bénilan, L. Boccardo, and M.A. Herrero, “On the limit of solutions of
*u*_{t}= Δ*u*^{m}as*m*→ ∞”,*Rend. Sem. Math. Univ. Politecn. (Torino)*, Fasc. spec., 1–13 (1989).Google Scholar - 42.A. S. Kalashnikov, “Perturbation of critical exponents in some nonlinear problems of mathematical physics,”
*Dokl. Akad. Nauk*,**337**No. 3, 320–322 (1994).MATHGoogle Scholar - 43.D. G. Aronson, M. G. Crandall, and L. A. Peletier, “Stabilization of solutions of a degenerate nonlinear diffusion problem,”
*Nonlin Anal.*,**6**, No. 10, 1001–1022 (1982).MathSciNetGoogle Scholar - 44.G. I. Barenblatt, “On some unsteady motions of fluids and gases in porous media,”
*Prikl. Mat. Mekh.*,**16**, No. 1, 67–78 (1952).MATHMathSciNetGoogle Scholar