PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
1999 | 81 | 1 | 101-122
Tytuł artykułu

Nonlinear Heat Equation with a Fractional Laplacian in a Disk

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
For the nonlinear heat equation with a fractional Laplacian $u_t + (-Δ)^{α/2} u = u^2$, 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution is obtained.
Rocznik
Tom
81
Numer
1
Strony
101-122
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-11-06
poprawiono
1999-02-01
Twórcy
  • Departamento de Matemáticas, Escuela Colombiana de Ingeniería, A.A. 14520, Bogotá, Colombia
Bibliografia
  • [1] C. Bardos, P. Penel, U. Frisch and P. Sulem, Modified dissipativity for a nonlinear evolution equation arising in turbulence, Arch. Rational Mech. Anal. 71 (1979), 237-256.
  • [2] P. Biler, T. Funaki and W. Woyczynski, Fractal Burgers equations, J. Differential Equations 148 (1998), 9-46.
  • [3] P. Biler, G. Karch and W. Woyczynski, Asymptotics for multifractal conservation laws, Studia Math. 135 (1999), 231-252.
  • [4] H. Brezis, Semilinear equations in $R^N$ without conditions at infinity, Appl. Math. Optim. 12 (1984), 271-282.
  • [5] H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983), 73-97.
  • [6] M. Escobedo, O. Kavian and H. Matano, Large time behavior of solutions of a dissipative semilinear heat equation, Comm. Partial Differential Equations 20 (1995), 1427-1452.
  • [7] A. Gmira and L. Veron, Large time behaviour of the solutions of a semilinear parabolic equation in $ℝ^N$, J. Differential Equations 53 (1984), 258-276.
  • [8] L. Herraiz, Asymptotic behavior of solutions of some semilinear parabolic problems, Ann. Inst. Henri Poincaré Anal. Non Linéaire 16 (1999), 49-105.
  • [9] L. Herraiz, A nonlinear parabolic problem in an exterior domain, J. Differential Equations 142 (1998), 371-412.
  • [10] E. Jahnke, F. Emde and F. Lösch, Tables of Higher Functions, 6th ed., Teubner, Stuttgart, 1960.
  • [11] S. Kamin and L. A. Peletier, Large time behaviour of solutions of the heat equation with absorption, Ann. Scuola Norm. Sup. Pisa 12 (1985), 393-408.
  • [12] S. Kamin and M. Ughi, On the behaviour as t → ∞ of the solutions of the Cauchy problem for certain nonlinear parabolic equations, J. Math. Anal. Appl. 128 (1987), 456-469.
  • [13] J. A. Mann, Jr. and W. Woyczynski, Rough surfaces generated by nonlinear transport, invited paper, Symposium on Nonlinear Diffusion, TMS International Meeting, September 1997.
  • [14] P. I. Naumkin and I. A. Shishmarëv, Nonlinear Nonlocal Equations in the Theory of Waves, Transl. Math. Monographs 133, Amer. Math. Soc., Providence, 1994.
  • [15] L. Oswald, Isolated positive singularities for a nonlinear heat equation, Houston J. Math. 14 (1988), 543-572.
  • [16] A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: solutions and applications, Chaos 7 (1997), 753-764.
  • [17] N. Sugimoto, 'Generalized' Burgers equations and fractional calculus, in: Nonlinear Wave Motion, A. Jeffrey (ed.), Longman Sci., Harlow, 1989, 162-179.
  • [18] G. Tolstov, Fourier Series, Dover, New York, 1962.
  • [19] V. V. Varlamov, On the Cauchy problem for the damped Boussinesq equation, Differential Integral Equations 9 (1996), 619-634.
  • [20] V. V. Varlamov, On the initial-boundary value problm for the damped Boussinesq equation, Discrete Contin. Dynam. Systems 4 (1998), 431-444.
  • [21] V. V. Varlamov, Long-time asymptotics of solutions of the second initial-boundary value problem for the damped Boussinesq equation, Abstract Appl. Anal., 2 (1998), 97-115.
  • [22] V. V. Varlamov, On the damped Boussinesq equation in a circle, Nonlinear Anal., to appear.
  • [23] V. V. Varlamov, On the spatially two-dimensional Boussinesq equation in a circular domain, ibid., submitted.
  • [24] M. Vishik, Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge Univ. Press, Cambridge, 1992.
  • [25] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed., Cambridge Univ. Press, London, 1966.
  • [26] C. E. Wayne, Invariant manifolds for parabolic partial differential equations on unbounded domains, Arch. Rational Mech. Anal. 138 (1997), 279-306.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv81i1p101bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.