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Non-negative solutions of the evolution p-Laplacian equation. Initial traces and Cauchy problem when \(1<p<2\). (English) Zbl 0726.35066

Die Autoren studieren nichtnegative Lösungen der Gleichung \[ u_ t- div(| Du|^{p-2}Du)=0 \text{für }x\in {\mathbb{R}}^ N,\quad t\in (0,T)\text{ und }1<p<2. \] Verschiedene schwache Lösungsbegriffe werden diskutiert. Die Lösungen sind i.a. nicht in \(L^{\infty}_{loc}\). Dennoch wird eine Ungleichung vom Harnackschen Typ hergeleitet und eine Existenz- und Eindeutigkeitstheorie für das Cauchy-Problem entwickelt.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35B45 A priori estimates in context of PDEs
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[1] S. N. Antonsev, Axially symmetric problems of gas dynamics with free boundaries, Doklady Akad. Nauk SSSR 216 (1974), pp. 473-476.
[2] D. G. Aronson & L. A. Caffarelli, The initial trace of a solution of the porous medium equation, Trans. AMS 280 (1983), pp. 351-366. · Zbl 0556.76084 · doi:10.1090/S0002-9947-1983-0712265-1
[3] P. Baras & M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), pp. 185-206. · Zbl 0519.35002
[4] P. Baras & M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal. 18 (1984), pp. 111-149. · Zbl 0582.35060 · doi:10.1080/00036818408839514
[5] P. Benilan & M. G. Crandall, Regularizing effects of homogeneous evolution equations, MRC Tech. Rep. # 2076, Madison Wi. (1980).
[6] P. Benilan, M. G. Crandall & M. Pierre, Solutions of the porous medium medium equation in R N under optimal conditions on initial values, Indiana Univ. Math. Jour. 33 (1984), pp. 51-87. · Zbl 0552.35045 · doi:10.1512/iumj.1984.33.33003
[7] L. Boccardo & T. Gallouët, Non linear elliptic and parabolic equations involving measure data, J. Funct. Anal. (to appear). · Zbl 0707.35060
[8] H. Brezis & A. Friedman, Non linear parabolic equations involving measures as initial conditions, J. Math. Pures et Appl. 62 (1983), pp. 73-97. · Zbl 0527.35043
[9] B. E. J. Dahlberg & C. E. Kenig, Non negative solutions of generalized porous medium equations, Revista Matematica Iberoamericana 2 (1986), pp. 267-305. · Zbl 0644.35057
[10] E. Di Benedetto, C 1,? local regularity of weak solutions of degenerate elliptic equations, Non Linear Anal. TMA 7 (1983), pp. 827-850. · Zbl 0539.35027 · doi:10.1016/0362-546X(83)90061-5
[11] E. Di Benedetto & M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation, Trans. Amer. Math. Soc. 314 (1989), pp. 187-224. · Zbl 0691.35047
[12] E. Di Benedetto & A. Friedman, Hölder estimates for non linear degenerate parabolic systems, Jour, für die Reine und Angewandte Math. 357 (1985), pp. 1-22.
[13] E. Di Benedetto & Chen Ya-zhe, On the local behavior of solutions of singular parabolic equations, Archive for Rational Mech. Anal. 103 (1988), pp. 319-346.
[14] E. Di Benedetto & Chen Ya-zhe, Boundary estimates for solutions of non linear degenerate parabolic systems, Jour, für die Reine und Angewandte Math. 395 (1989), pp. 102-131.
[15] L. C. Evans, Application of non linear semigroup theory to certain partial differential equations, in Non Linear evolution Equations, M. G. Crandall Editor (1979).
[16] M. A. Herrero & J. L. Vazquez, Asymptotic behaviour of the solutions of a strongly non linear parabolic problem, Ann. Faculté des Sciences Toulouse 3 (1981), pp. 113-127. · Zbl 0498.35013
[17] M. A. Herrero & M. Pierre, The Cauchy problem for u t=?(u m) when 0<m< 1, Trans. AMS 291 (1985), pp. 145-158. · Zbl 0583.35052
[18] L. I. Kamynin, The existence of solutions of Cauchy problems and boundary-value problems for a second order parabolic equation in unbounded domains: I, Differential Equations 23 (1987), pp. 1315-1323. · Zbl 0653.35033
[19] O. A. Ladyzhenskaya, N. A. Solonnikov, & N. N. Ural’tzeva, Linear and quasi linear equations of parabolic type, Trans. Math. Mono. # 23 AMS Providence R.I. (1968).
[20] O. A. Ladyzenskajia, New equations for the description of motion of viscous incompressible fluids and solvability in the large of boundary value problems for them, Proc. Steklov Inst. Math. # 102 (1967), pp. 95-118 (transl. Trud. Trudy Math. Inst. Steklov # 102 (1967), pp. 85-104).
[21] J. L. Lions, Quelques méthodes de resolution des problèmes aux limites non linéaires, Dunod, Paris (1969).
[22] L. K. Martinson & K. B. Paplov, Unsteady shear flows of a conducting fluid with a rheological power law, Magnit. Gidrodinamika 2 (1970), pp. 50-58.
[23] L. K. Martinson & K. B. Paplov, The effect of magnetic plasticity in non-Newtonian fluids, Magnit. Gidrodinamika 3 (1969), pp. 69-75.
[24] G. Minty, Monotone (non linear) operators in Hilbert spaces, Duke Math. J. 29 (1967), pp. 341-346. · Zbl 0111.31202 · doi:10.1215/S0012-7094-62-02933-2
[25] L. E. Payne & G. A. Philippin, Some applications of the maximum principle in the problem of torsional creep, SIAM Jour. Math. Anal. 33 (1977), pp. 446-455. · Zbl 0378.73028 · doi:10.1137/0133028
[26] M. Pierre, Non linear fast diffusion with measures as data, Proceedings of Non linear parabolic equations: Qualitative properties of solutions, Tesei & Boccardo Eds. Pitman # 149 (1985).
[27] M. Pierre, Uniqueness of the solutions of u t-?(u) m=0 with initial datum a measure, Non Lin. Anal. TMA, 6 (1982), pp. 175-187. · Zbl 0484.35044 · doi:10.1016/0362-546X(82)90086-4
[28] E. S. Sabinina, A class of nonlinear degenerate parabolic equations, Sov. Math. Doklady # 143 (1962), pp. 495-498. · Zbl 0122.33503
[29] S. Tacklind, Sur les classes quasianalitiques des solutions des équations aux dérivées partielles du type parabolique, Acta Reg. Soc. Sc. Uppsaliensis (Ser. 4) 10 (1936), pp. 3-55.
[30] A. N. Tychonov, Théorèmes d’unicité pour l’équation de la chaleur, Math. Sbornik 42 (1935), pp. 199-216.
[31] D. V. Widder, Positive temperatures in an infinite rod, Trans. AMS 55 (1944), pp. 85-95. · Zbl 0061.22303
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