Journal of Dynamics and Differential Equations

, Volume 19, Issue 4, pp 985–1005 | Cite as

Localized Non-diffusive Asymptotic Patterns for Nonlinear Parabolic Equations with Gradient Absorption

We study the large-time behaviour of the solutions u of the evolution equation involving nonlinear diffusion and gradient absorption
$$\partial_t u - \Delta_p u + \vert\nabla u\vert^q = 0$$
We consider the problem posed for \(x \in \mathbb R^N \) and t  >  0 with non-negative and compactly supported initial data. We take the exponent p  >  2 which corresponds to slow p-Laplacian diffusion, and the exponent q in the superlinear range 1  <  q  <  p  −  1. In this range the influence of the Hamilton–Jacobi term \( \vert\nabla u\vert^q\) is determinant, and gives rise to the phenomenon of localization. The large-time behaviour is described in terms of a suitable self-similar solution that solves a Hamilton–Jacobi equation. The shape of the corresponding spatial pattern is rather conical instead of bell-shaped or parabolic.


Nonlinear parabolic equations p-Laplacian equation asymptotic patterns localization Hamilton–Jacobi equations viscosity solutions 

AMS Subject Classification

35B40 35K65 49L25 


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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institut de Mathématiques de Toulouse, CNRS UMR 5219Université Paul Sabatier (Toulouse III)Toulouse Cedex 9France
  2. 2.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

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