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

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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.

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Authors and Affiliations

  1. Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université Paul Sabatier (Toulouse III), 118 route de Narbonne, F-31062, Toulouse Cedex 9, France

    Philippe Laurençot

  2. Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, E-28049, Madrid, Spain

    Juan Luis Vázquez

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  1. Philippe Laurençot
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  2. Juan Luis Vázquez
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Correspondence to Philippe Laurençot.

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Dedicated to Pavol Brunovský.

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Laurençot, P., Vázquez, J.L. Localized Non-diffusive Asymptotic Patterns for Nonlinear Parabolic Equations with Gradient Absorption. J Dyn Diff Equat 19, 985–1005 (2007). https://doi.org/10.1007/s10884-007-9093-y

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  • DOI: https://doi.org/10.1007/s10884-007-9093-y

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