Lyapunov functionals in singular limits for perturbed quasilinear degenerate parabolic equations

  1. Chaves, Manuela 3
  2. Galaktionov, Victor A. 12
  1. 1 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK
  2. 2 Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, 125047 Moscow, Russia
  3. 3 Department of Mathematics, Autonoma University of Madrid, 28049 Madrid, Spain
Mostrar afiliaciones +
Revista:
Analysis and Applications

ISSN: 0219-5305,  1793-6861

Año de publicación: 2003

Volumen: 1

Número: 4

Páginas: 351-385

Tipo: Artículo

DOI: 10.1142/S0219530503000193 GOOGLE SCHOLAR lock_openAcceso abierto editor

Otras publicaciones en: Analysis and Applications

Resumen

As a key example, we study the asymptotic behaviour near finite focusing time t=T of radial solutions of the porous medium equation with absorption with bounded compactly supported initial data u(x,0)=u0(|x|), and exponents m>1 and p>pc, where pc=pc(m,N)∈(-m,0) is a critical exponent. We show that under certain assumptions, the behaviour of the solution as t→T- near the origin is described by self-similar Graveleau solutions of the porous medium equation ut=Δum. In the rescaled variables, we deal with an exponential non-autonomous perturbation of a quasilinear parabolic equation, which is shown to admit an approximate Lyapunov functional. The result is optimal, and in the critical case p=pc an extra ln(T-t) scaling of the Graveleau asymptotics is shown to occur. Other types of self-similar and non self-similar focusing patterns are discussed

Referencias bibliográficas

  • Angenent S., J. London Math. Soc. (2), 42, pp. 339
  • 10.1080/03605309508821130
  • 10.1137/0117045
  • Aronson D. G., Euro. J. Appl. Math., 4, pp. 65, 10.1017/S095679250000098X
  • Barenblatt G. I., Prikl. Mat. Mekh., 16, pp. 67
  • Barenblatt G. I., (1978), Similarity, Selfsimilarity and Intermediate Asymptotics
  • 10.1007/978-1-4612-4546-9
  • 10.1137/0523045
  • 10.1512/iumj.1984.33.33003
  • Birman M. S., (1987), Spectral Theory of Self-Adjoint Operators in Hilbert Space, 10.1007/978-94-009-4586-9
  • 10.1016/0022-0396(92)90104-U
  • 10.1098/rspa.1998.0263
  • 10.1007/978-1-4612-0895-2
  • 10.1137/S0036141093223419
  • 10.1016/0041-5553(86)90007-8
  • 10.1070/SM1986v054n02ABEH002979
  • 10.1007/s002050050048
  • Galaktionov V. A., Proc. Roy. Soc. Edinburgh, 125, pp. 877, 10.1017/S0308210500022538
  • 10.1142/S0219199799000031
  • 10.1016/0022-1236(91)90120-T
  • 10.1080/03605309408821047
  • 10.1007/BF00383674
  • 10.1006/jdeq.1996.0059
  • 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.0.CO;2-H
  • 10.1002/cpa.3160380304
  • Herrero M. A., Ann. Inst. Henri Poincaré, Analyse non linéaire, 10, pp. 131, 10.1016/S0294-1449(16)30217-7
  • Kalashnikov A. S., Moscow Univ. Math. Bull., 29, pp. 48
  • 10.1070/RM1987v042n02ABEH001309
  • Kolmogorov A. N., Byull. Moskov. Gos. Univ., Sect. A, 1, pp. 1
  • 10.1016/0362-546X(85)90057-4
  • 10.1007/978-94-011-3748-5
  • 10.1007/978-3-0348-9234-6
  • Matano H., J. Math. Kyoto Univ. (JMKYAZ), 18, pp. 221, 10.1215/kjm/1250522572
  • 10.1002/(SICI)1097-0312(199802)51:2<139::AID-CPA2>3.0.CO;2-C
  • 10.1070/RM1961v016n05ABEH004114
  • 10.1007/978-1-4612-5561-1
  • Pelcé P., (1988), Dynamics of Curved Fronts
  • Samarskii A. A., (1987), Blow-up in Quasilinear Parabolic Equations
  • Sturm C., J. Math. Pures Appl., 1, pp. 373
  • Vazquez J. L., Nonlinear Anal., TMA, 15, pp. 932
  • Zelenyak T. I., Differential Equation, 4, pp. 17