Skip to main content
Log in

Non-negative solutions of the evolution p-Laplacian equation. Initial traces and cauchy problem when 1<p<2

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

References

  1. S. N. Antonsev, Axially symmetric problems of gas dynamics with free boundaries, Doklady Akad. Nauk SSSR 216 (1974), pp. 473–476.

    Google Scholar 

  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.

    Google Scholar 

  3. P. Baras & M. Pierre, Singularités éliminables pour des équations semi-linéaires, Ann. Inst. Fourier (Grenoble) 34 (1984), pp. 185–206.

    Google Scholar 

  4. P. Baras & M. Pierre, Problèmes paraboliques semi-linéaires avec données mesures, Applicable Anal. 18 (1984), pp. 111–149.

    Google Scholar 

  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.

    Google Scholar 

  7. L. Boccardo & T. Gallouët, Non linear elliptic and parabolic equations involving measure data, J. Funct. Anal. (to appear).

  8. H. Brezis & A. Friedman, Non linear parabolic equations involving measures as initial conditions, J. Math. Pures et Appl. 62 (1983), pp. 73–97.

    Google Scholar 

  9. B. E. J. Dahlberg & C. E. Kenig, Non negative solutions of generalized porous medium equations, Revista Matematica Iberoamericana 2 (1986), pp. 267–305.

    Google Scholar 

  10. E. Di Benedetto, C 1,ά local regularity of weak solutions of degenerate elliptic equations, Non Linear Anal. TMA 7 (1983), pp. 827–850.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  17. M. A. Herrero & M. Pierre, The Cauchy problem for u t=Δ(u m) when 0<m< 1, Trans. AMS 291 (1985), pp. 145–158.

    Google Scholar 

  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.

    Google Scholar 

  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).

    Google Scholar 

  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).

    Google Scholar 

  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.

    Google Scholar 

  23. L. K. Martinson & K. B. Paplov, The effect of magnetic plasticity in non-Newtonian fluids, Magnit. Gidrodinamika 3 (1969), pp. 69–75.

    Google Scholar 

  24. G. Minty, Monotone (non linear) operators in Hilbert spaces, Duke Math. J. 29 (1967), pp. 341–346.

    Google Scholar 

  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.

    Google Scholar 

  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.

    Google Scholar 

  28. E. S. Sabinina, A class of nonlinear degenerate parabolic equations, Sov. Math. Doklady # 143 (1962), pp. 495–498.

  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.

    Google Scholar 

  30. A. N. Tychonov, Théorèmes d'unicité pour l'équation de la chaleur, Math. Sbornik 42 (1935), pp. 199–216.

    Google Scholar 

  31. D. V. Widder, Positive temperatures in an infinite rod, Trans. AMS 55 (1944), pp. 85–95.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Additional information

Communicated by H. Brezis

Rights and permissions

Reprints and permissions

About this article

Cite this article

Di Benedetto, E., Herrero, M.A. Non-negative solutions of the evolution p-Laplacian equation. Initial traces and cauchy problem when 1<p<2. Arch. Rational Mech. Anal. 111, 225–290 (1990). https://doi.org/10.1007/BF00400111

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00400111

Keywords

Navigation