Complutense University Library

Approaching a vertex in a shrinking domain under a nonlinear flow


Herrero, Miguel A. and Ughi, M. and Velázquez, J.J. L. (2004) Approaching a vertex in a shrinking domain under a nonlinear flow. NoDea-Nonlinear Differential Equations and Applications, 11 . pp. 1-28. ISSN 1021-9722

[img] PDF
Restringido a Repository staff only hasta 31 December 2020.


Official URL:


We consider here the homogeneous Dirichlet problem for the equation u(t)= uΔu - γ|∇u|(2) with γ ∈ R, u ≥ 0, in a noncylindrical domain in space-time given by |x| ≤ R(t) = (T - t)(p), with p > 0. By means of matched asymptotic expansion techniques we describe the asymptotics of the maximal solution approaching the vertex x = 0, t = T, in the three different cases p > 1/2, p = 1/2(vertex regular), p < 1/2 (vertex irregular).

Item Type:Article
Uncontrolled Keywords:Asymptotics; nonlinear flow; degenerate parabolic equation; viscosity solutions; Dirichlet problem; heat-equation; singularities; regularity; points
Subjects:Sciences > Physics > Mathematical physics
Sciences > Mathematics > Differential equations
ID Code:16426

L.J.S. ALLEN, Persistence and extinction in single-species reaction-diffusion models. Bull. Math. Biology 45 (2) (1983), 209–227.

D.G. ARONSON, The porous medium equation, in Nonlinear diffusion problems (A. Fasano and M. Primicerio eds.), Lecture Notes in Math. 1224, Springer (1985), 1–46.

M. ABRAMOWITZ, I. STEGUN, Handbook of mathematical functions. Dover 1964.

S.B. ANGENENT, J.J.L. VELÁZQUEZ, Asymptotic shape of cusp singularities in curve shortening. Duke Math. J. 77 (1) (1995), 71–110.

G.I. BARENBLATT, Scaling, self-similarity, and intermediate asymptotics. Cambridge Texts in Applied Mathematics Vol. 14 (1996).

G.I. BARENBLATT, M. BERTSCH, A.E. CHERTOCK, V.M. PROSTOKINSHIN, Self-similar intermediate asymptotics for a degenerate parabolic filtration-absorption equation. Submitted to Proc. Nat. Acad. Sci. (USA), 2000.

M. BERTSCH, R. DAL PASSO, M. UGHI, Discontinuous viscosity solutions of a degenerate parabolic equation. Trans. Amer. Math. Soc. 320 (2) (1990), 779–798.

M. BERTSCH, R. DAL PASSO, M. UGHI, Nonuniqueness of solutions of a degenerate parabolic equation. Annali Mat. Pura et Appl. CLXI (1992), 57–81.

C.M. BENDER, S.A. ORSZAG, Advanced mathematical methods for scientists and engineers. Mc Graw Hill 1978.

M. BERTSCH, M. UGHI, Positivity properties of viscosity solutions of a degenerate parabolic equation. J. Nonl. Anal. 14 (7) (1990), 571–592.

R. DAL PASSO, S. LUCKHAUS, On a degenerate diffusion problem not in divergence form. J. Diff. Eq. 69 (1987), 1–14.

R. DAL PASSO, M. UGHI, Problème de Dirichlet pour une classe d’equations paraboliques non lináires dégenerées dans des ouverts non cylindriques. C. R. Acad. Sci. Paris t. 308, Series I (1989), 555–558.

R. DAL PASSO, M. UGHI, On a class of nonlinear degenerate parabolic equations in non-cylindrical domains: existence and regularity. Rendiconti di Matematica, Università di Roma la Sapienza, 57 (9) (1989), 445–456.

L.C. EVANS, R. GARIEPY, Wiener’s criterion for the heat equation. Arch. Rat. Mech. Anal. 78 (4) (1982), 293–314.

M. ESCOBEDO, M.A. HERRERO, J.J.L. VELÁZQUEZ, A nonlinear Fokker-Planck equation modelling the approach to thermal equilibrium in a homogeneous plasma. Trans. Amer. Math. Soc. 350 (10) (1998), 3837–3901.

E.G. EFFROS, J.L. KAZDAN, Applications of Choquet simplexes to elliptic and parabolic boundary problems. J. Diff. Eq. 8 (1970), 95–134.

E.G. EFFROS, J.L. KAZDAN, On the Dirichlet problem for the heat equation. Indiana U. Math. J. 20 (1971), 683–693.

A. FRIEDMAN, J.B. MC LEOD, Blow-up of solutions of nonlinear degenerate parabolic equations. Arch. Rat. Mech. Anal. (1986), 55–80.

I. FUKUDA, H. ISHII, M. TSUTSUMI, Uniqueness of solutions to the Cauchy problem for ut = uΔu − γ|∇u|2. Diff. Int. Equ. 6 (1993), 1231–1252.

M.A. HERRERO, J.J.L. VELÁZQUEZ, Singularity patterns in a chemotaxis model. Math. Annalen 306 (3) (1996), 583–623.

M.A.HERRERO, A.A. LACEY, J.J.L. VELÁZQUEZ, Blow-up under oscillatory boundary conditions. Asymptotic Analysis 9 (1994), 1–22.

O.D. KELLOGG, Foundations of potential theory. Dover 1954.

B.C. LOW, Resistive diffusion of force-free magnetic fields in a passive medium. Astrophysical Journal 181 (1973), 917–929.

J.D. MURRAY, Mathematical Biology, Springer 1993.

J. MOSSINO, M. UGHI, Isoperimetric inequalities and regularity at shrinking points for parabolic problems. J. Nonl. Anal. 39 (2000), 499–517.

J. MOSSINO, M. UGHI, On the regularity at shrinking points for the porous media equation. Nonl. Diff. Eq. and Appl. 7 (2000), 1–21.

I.G. PETROVSKY, Zur ersten Randwertaufgabe der Wärmeleitungsgleichung. Compositio Math. 1 (1935), 383–419.

M. UGHI, A degenerate parabolic equation modelling the spread of a epidemic. Ann. Mat. Pura Appl. 143 (1986), 385–400.

J.J.L. VELÁZQUEZ, Classification of singularities for blowing up solutions in higher dimensions. Trans. Amer. Math. Soc. 338 (1) (1993), 441–464.

Deposited On:19 Sep 2012 08:16
Last Modified:07 Feb 2014 09:29

Repository Staff Only: item control page