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Singular behavior in nonlinear parabolic equations. (English) Zbl 0573.35046
At the beginning the authors study the question of singular solutions for elliptic boundary value problems \(\Delta u+f(u)=0\) on \(\Omega\), \(u=0\) on \(\partial \Omega\), \(u>0\) on \(\Omega\). Singular solution is the \(C^ 2(\Omega \setminus \{0\})\) solution of the problem for which, \(\overline{\lim}_{x\to 0}u(x)=+\infty.\) In theorem 1 the conditions on f (general) are formulated under which the problem has (i) infinitely many singular solutions, (ii) no singular solution. Theorem 2 contains the result for the special case \(f=\lambda | x|^{\ell} u^ p.\)
Further, the boundary value problems with the conditions \(u(x,t)=0\), (x,t)\(\in \partial \Omega\), \(t>0\) and \(u(x,0)=u^ 0(x)\), \(x\in \Omega\) for either the equation (1) \(u_ t=\Delta u+u^ p\), or the equation (2) \(u_ t=\Delta (u^ m)\) are considered. In theorem 3, which concerns the equation (1), the conditions on p are given guaranteeing the existence of \(u^ 0\in L_ p(\Omega)\) such that the problem has at least two solutions from \(C([0,T];L_ p(\Omega))\). The asymptotic behaviour of the non uniformly bounded solutions of this problems is established in theorem 4.
Finally for the problem of ”fast diffusion” (equation (2)) there are given the conditions under which for each \(t_ 0>0\) there is a solution with \(u_ 0\in L_ q(\Omega)\) and \(u(.,t_ 0)\not\in L^{\infty}(\Omega)\).
Reviewer: O.John

MSC:
35K55 Nonlinear parabolic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J60 Nonlinear elliptic equations
35R25 Ill-posed problems for PDEs
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[1] Antonio Ambrosetti and Paul H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Functional Analysis 14 (1973), 349 – 381. · Zbl 0273.49063
[2] Patricio Aviles, On isolated singularities in some nonlinear partial differential equations, Indiana Univ. Math. J. 32 (1983), no. 5, 773 – 791. · Zbl 0548.35042 · doi:10.1512/iumj.1983.32.32051 · doi.org
[3] P. Baras, Non-unicité des solutions d’une equation d’evolution non-lineaire, preprint.
[4] P. Bénilan, Opérateurs accrétifs et semigroups dans les espaces \( {L^p}\;(1 \leqslant p \leqslant \infty )\), France-Japan Seminar, Tokyo, 1976.
[5] Philippe Bénilan and J. Ildefonso Díaz, Comparison of solutions of nonlinear evolution problems with different nonlinear terms, Israel J. Math. 42 (1982), no. 3, 241 – 257. · Zbl 0506.35049 · doi:10.1007/BF02802726 · doi.org
[6] James G. Berryman and Charles J. Holland, Stability of the separable solution for fast diffusion, Arch. Rational Mech. Anal. 74 (1980), no. 4, 379 – 388. · Zbl 0458.35046 · doi:10.1007/BF00249681 · doi.org
[7] Haïm Brézis and Avner Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. (9) 62 (1983), no. 1, 73 – 97. · Zbl 0527.35043
[8] Haïm Brézis and Pierre-Louis Lions, A note on isolated singularities for linear elliptic equations, Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 263 – 266. · Zbl 0468.35036
[9] Haïm Brézis and Louis Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (1983), no. 4, 437 – 477. · Zbl 0541.35029 · doi:10.1002/cpa.3160360405 · doi.org
[10] S. Chandrasekhar, An introduction to the study of stellar structure, Dover Publications, Inc., New York, N. Y., 1957. · Zbl 0079.23901
[11] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), no. 3, 209 – 243. · Zbl 0425.35020
[12] -, Symmetry of positive solutions of nonlinear elliptic equations in \( {{\mathbf{R}}^n}\), Adv. in Math. Suppl. Stud., Vol. 7A, Academic Press, New York and London, 1981, pp. 369-402.
[13] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no. 4, 525 – 598. · Zbl 0465.35003 · doi:10.1002/cpa.3160340406 · doi.org
[14] D. Gilbarg and James Serrin, On isolated singularities of solutions of second order elliptic differential equations, J. Analyse Math. 4 (1955/56), 309 – 340. · Zbl 0071.09701 · doi:10.1007/BF02787726 · doi.org
[15] Alain Haraux and Fred B. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982), no. 2, 167 – 189. · Zbl 0465.35049 · doi:10.1512/iumj.1982.31.31016 · doi.org
[16] M. Hénon, Numerical experiments on the stability of spherical stellar systems, Astronom. and Astrophys. 24 (1973), 229-238.
[17] D. D. Joseph and T. S. Lundgren, Quasilinear Dirichlet problems driven by positive sources, Arch. Rational Mech. Anal. 49 (1972/73), 241 – 269. · Zbl 0266.34021 · doi:10.1007/BF00250508 · doi.org
[18] Stanley Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963), 305 – 330. · Zbl 0156.33503 · doi:10.1002/cpa.3160160307 · doi.org
[19] Линейные и квазилинейные уравнения параболического типа, Издат. ”Наука”, Мосцощ, 1967 (Руссиан). О. А. Ладыžенскаја, В. А. Солонников, анд Н. Н. Урал\(^{\приме}\)цева, Линеар анд чуасилинеар ечуатионс оф параболиц тыпе, Транслатед фром тхе Руссиан бы С. Смитх. Транслатионс оф Матхематицал Монограпхс, Вол. 23, Америцан Матхематицал Социеты, Провиденце, Р.И., 1968 (Руссиан).
[20] Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. · Zbl 0164.13002
[21] Howard A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form \?\?_\?=-\?\?+\Cal F(\?), Arch. Rational Mech. Anal. 51 (1973), 371 – 386. · Zbl 0278.35052 · doi:10.1007/BF00263041 · doi.org
[22] P.-L. Lions, Isolated singularities in semilinear problems, J. Differential Equations 38 (1980), no. 3, 441 – 450. · Zbl 0458.35033 · doi:10.1016/0022-0396(80)90018-2 · doi.org
[23] Wei-Ming Ni, Uniqueness, nonuniqueness and related questions of nonlinear elliptic and parabolic equations, Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983) Proc. Sympos. Pure Math., vol. 45, Amer. Math. Soc., Providence, RI, 1986, pp. 229 – 241.
[24] Wei Ming Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J. 31 (1982), no. 6, 801 – 807. · Zbl 0515.35033 · doi:10.1512/iumj.1982.31.31056 · doi.org
[25] Wei-Ming Ni, Uniqueness of solutions of nonlinear Dirichlet problems, J. Differential Equations 50 (1983), no. 2, 289 – 304. · Zbl 0476.35033 · doi:10.1016/0022-0396(83)90079-7 · doi.org
[26] Wei-Ming Ni and Roger D. Nussbaum, Uniqueness and nonuniqueness for positive radial solutions of \Delta \?+\?(\?,\?)=0, Comm. Pure Appl. Math. 38 (1985), no. 1, 67 – 108. · Zbl 0581.35021 · doi:10.1002/cpa.3160380105 · doi.org
[27] Wei-Ming Ni and Paul Sacks, The number of peaks of positive solutions of semilinear parabolic equations, SIAM J. Math. Anal. 16 (1985), no. 3, 460 – 471. · Zbl 0605.35041 · doi:10.1137/0516033 · doi.org
[28] Wei-Ming Ni, Paul E. Sacks, and John Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations, J. Differential Equations 54 (1984), no. 1, 97 – 120. · Zbl 0565.35053 · doi:10.1016/0022-0396(84)90145-1 · doi.org
[29] L. Peletier, The porous medium equation, Application of Nonlinear Analysis in the Physical Sciences, Pitman, New York, 1981. · Zbl 0497.76083
[30] S. Pohozaev, Eigenfunctions of the equation \( \Delta u + \lambda f(u) = 0\), Soviet Math. J. 6 (1965), 1408-1411. · Zbl 0141.30202
[31] James Serrin, Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247 – 302. · Zbl 0128.09101 · doi:10.1007/BF02391014 · doi.org
[32] Laurent Véron, Effets régularisants de semi-groupes non linéaires dans des espaces de Banach, Ann. Fac. Sci. Toulouse Math. (5) 1 (1979), no. 2, 171 – 200 (French, with English summary). · Zbl 0426.35052
[33] Fred B. Weissler, Local existence and nonexistence for semilinear parabolic equations in \?^\?, Indiana Univ. Math. J. 29 (1980), no. 1, 79 – 102. · Zbl 0443.35034 · doi:10.1512/iumj.1980.29.29007 · doi.org
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