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Estimates of Solutions and Asymptotic Symmetry for Parabolic Equations on Bounded Domains

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Abstract

We consider fully nonlinear parabolic equations on bounded domains under Dirichlet boundary conditions. Assuming that the equation and the domain satisfy certain symmetry conditions, we prove that each bounded positive solution of the Dirichlet problem is asymptotically symmetric. Compared with previous results of this type, we do not assume certain crucial hypotheses, such as uniform (with respect to time) positivity of the solution or regularity of the nonlinearity in time. Our method is based on estimates of solutions of linear parabolic problems, in particular on a theorem on asymptotic positivity of such solutions.

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References

  1. Alexandrov A.D. A characteristic property of spheres. Ann. Math. Pura Appl. (4) 58, 303–354 (1962)

    Google Scholar 

  2. Babin A.V. (1994) Symmetrization properties of parabolic equations in symmetric domains. J. Dynam. Differential Equations 6, 639–658

    Article  MATH  MathSciNet  ADS  Google Scholar 

  3. Babin A.V. (1995) Symmetry of instabilities for scalar equations in symmetric domains. J. Differential Equations 123, 122–152

    Article  MATH  MathSciNet  Google Scholar 

  4. Babin A.V., Sell G.R. (2000) Attractors of non-autonomous parabolic equations and their symmetry properties. J. Differential Equations 160, 1–50

    Article  MATH  MathSciNet  Google Scholar 

  5. Berestycki, H. Qualitative properties of positive solutions of elliptic equations. Partial Differential Equations (Praha, 1998), Research Notes in Mathematics volume 406, Chapman & Hall/CRC, Boca Raton, FL, pp. 34–44 2000

  6. Berestycki H., Nirenberg L. (1991) On the method of moving planes and the sliding  method. Bull. Braz. Math. Soc. (NS) 22, 1–37

    Article  MATH  MathSciNet  Google Scholar 

  7. Brunovský P., Fiedler B. (1986) Numbers of zeros on invariant manifolds in reaction-diffusion equations. Nonlinear Anal. 10, 179–193

    Article  MATH  MathSciNet  Google Scholar 

  8. Busca J., Jendoubi M.-A., PoláčIK P. (2002) Convergence to equilibrium for semilinear parabolic problems in \(\mathbb{R}\) N. Comm. Partial Differential Equations 27, 1793–1814

    Article  MATH  MathSciNet  Google Scholar 

  9. Busca J., Sirakov B. (2000) Symmetry results for semilinear elliptic systems in the whole  space. J. Differential Equations 163, 41–56

    Article  MATH  MathSciNet  Google Scholar 

  10. Chen X.-Y., Poláčik P. (1996) Asymptotic periodicity of positive solutions of reaction diffusion equations on a ball. J. Reine Angew. Math. 472, 17–51

    MATH  MathSciNet  Google Scholar 

  11. Cho, S. Thesis, University of Minnesota, 2005.

  12. Cho, S., Safonov, M.V. Hölder regularity of solutions to second order elliptic equations in non-smooth domains. preprint.

  13. Cortázar C., Del Pino M., Elgueta M. (1999) The problem of uniqueness of the limit in a semilinear heat equation. Comm. Partial Differential Equations 24, 2147–2172

    MATH  MathSciNet  Google Scholar 

  14. Da Lio, F., Sirakov, B. Symmetry results for viscosity solutions of fully nonlinear  elliptic equations. J. EUR. MATH. SOC., to appear

  15. Dancer E.N. (1992) Some notes on the method of moving planes. BULL. AUSTRAL. MATH. SOC. 46, 425–434

    Article  MATH  MathSciNet  Google Scholar 

  16. Dancer E.N., Hess P. (1994) The symmetry of positive solutions of periodic-parabolic  problems. J. Comput. Appl. Math. 52, 81–89

    Article  MATH  MathSciNet  Google Scholar 

  17. Evans L.C. (1998). Partial differential equations. Graduate Studies in Mathematics, Volume 19, American Mathematical Society, Providence, RI

    MATH  Google Scholar 

  18. Feireisl E., Petzeltová H. (1997) Convergence to a ground state as a threshold phenomenon in nonlinear parabolic equations. Differential Integral Equations 10, 181–196

    MATH  MathSciNet  Google Scholar 

  19. Gidas B., Ni W.-M., Nirenberg L. (1979). Symmetry and related properties via the  maximum principle. Comm. Math. Phys. 68, 209–243

    Article  MATH  MathSciNet  ADS  Google Scholar 

  20. Gidas B., Ni W.-M., Nirenberg L. (1981) Symmetry of positive solutions of nonlinear  elliptic equations in \(\mathbb{R}\) n. Mathematical Analysis and Applications, Part A, Academic Press, New York, pp. 369–402

    Google Scholar 

  21. Gruber, M. Harnack inequalities for solutions of general second order parabolic equations and estimates of their Hölder constants. Math. Z. 185, no. 1, 23–43 (1984)

  22. Haraux A., Poláčik P. (1992) Convergence to a positive equilibrium for some nonlinear evolution equations in a ball. Acta Math. Univ. Comenian. (N.S.) 61, 129–141

    MATH  MathSciNet  Google Scholar 

  23. Hess P., Poláčik P. (1994) Symmetry and convergence properties for non-negative solutions of nonautonomous reaction-diffusion problems. Proc. Roy. Soc. Edinburgh Sect. A 124, 573–587

    MATH  MathSciNet  Google Scholar 

  24. Húska, J. Exponential separation and principal Floquet bundles for linear parabolic equations on general bounded domains: divergence case. preprint.

  25. Húska, J., Poláčik, P., Safonov, M.V. Harnack inequalities, exponential separation, and perturbations of principal Floquet bundles for linear parabolic equations. preprint.

  26. Kawohl, B. Symmetrization–or how to prove symmetry of solutions to a PDE. Partial Differential Equations (Praha, 1998), Research Notes in Mathematics volume 406, Chapman & Hall/CRC, Boca Raton, FL, pp. 214–229 2000

  27. Krylov N.V. (1987) Nonlinear elliptic and parabolic equations of the second order. Mathematics and its Applications (Soviet Series), volume 7, D. Reidel Publishing Co., Dordrecht

    MATH  Google Scholar 

  28. Krylov N.V., Safonov M.V. (1980) Certain properties of solutions of parabolic equations with measurable coefficients. Izv. Akad. Nauk 44, 161–175

    MATH  MathSciNet  Google Scholar 

  29. Ladyzhenskaya O.A., Solonnikov V.A., Urall’ceva N.N. (1968) Linear and quasilinear equations of parabolic type. Translations of Mathematical Monographs, Volume. 23, American Mathematical Society, Providence, R.I.

    Google Scholar 

  30. Li C. (1991) Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains. Comm. Partial Differential Equations 16, 491–526

    MATH  MathSciNet  Google Scholar 

  31. Lieberman G.M. (1996) Second Order Parabolic Differential Equations. World Scientific Publishing Co. Inc., River Edge, NJ

    MATH  Google Scholar 

  32. Ni, W.-M. Qualitative properties of solutions to elliptic problems. Handbood of Differential Equations Stationary Partial Differential Equations, Volume 1 (M. Chipot and P. Quittner, eds.), Elsevier, pp. 157–233 2004

  33. Poláčik, P.: Symmetry properties of positive solutions of parabolic equations on \(\mathbb{R}\) N: I. Asymptotic symmetry for the Cauchy problem. Comm. Partial Differential Equations, to appear.

  34. Poláčik, P. Symmetry properties of positive solutions of parabolic equations on \(\mathbb{R}\) N: II. Entire solutions., Comm. Partial Differential Equations to appear.

  35. Roquejoffre J.-M. (1997) Eventual monotonicity and convergence to traveling fronts for the solutions of parabolic equations in cylinders. Ann. Inst. H. Poincaré Anal. Non Linéaire 14, 499–552

    Article  MATH  MathSciNet  ADS  Google Scholar 

  36. Serrin J. (1971) A symmetry problem in potential theory. Arch. Ration Mech. Anal. 43, 304–318

    Article  MATH  MathSciNet  Google Scholar 

  37. Serrin J., Zou H. (1999) Symmetry of ground states of quasilinear elliptic equations. Arch. Ration. Mech. Anal. 148, 265–290

    Article  MATH  MathSciNet  Google Scholar 

  38. Troy W.C. (1981) Symmetry properties in systems of semilinear elliptic equations. J. Differential Equations 42, 400–413

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA

    Peter Poláčik

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  1. Peter Poláčik
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Correspondence to Peter Poláčik.

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Communicated by V. Šverák

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Poláčik, P. Estimates of Solutions and Asymptotic Symmetry for Parabolic Equations on Bounded Domains. Arch Rational Mech Anal 183, 59–91 (2007). https://doi.org/10.1007/s00205-006-0004-x

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  • DOI: https://doi.org/10.1007/s00205-006-0004-x

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