Radial equivalence for the two basic nonlinear degenerate diffusion equations

Iagar, Razvan Gabriel; Sánchez, Ariel; Vázquez, Juan Luís

doi:10.1016/j.matpur.2007.09.002

Cited by 33 publications

(67 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Moreover, the connection between the free parameter C and the mass M C is given by For more details about the properties of the source-type solutions, the reader may consult the book [18]. On the other hand, it is proved in [11] that there exist many other self-similar solutions of the form…”

Section: Preliminariesmentioning

confidence: 99%

Asymptotic analysis of the p-Laplacian flow in an exterior domain

Iagar

Vázquez

2009

Ann. Inst. H. Poincaré Anal. Non Linéaire

Self Cite

View full text Add to dashboard Cite

We consider the Dirichlet problem for the p-Laplacian evolution equation, u t = p u, where p > 2, posed in an exterior domain in R N , with zero Dirichlet boundary condition and with integrable and nonnegative initial data. We are interested in describing the influence of the holes of the domain on the large time behaviour of the solutions. Such behaviour varies depending on the relative values of N and p. We must distinguish between the behaviour near infinity of space (outer analysis), and near the holes (inner analysis). We obtain that the outer analysis is given in all cases by certain self-similar solutions and the inner analysis is given by quasi-stationary states. Logarithmic corrections to exact self-similarity appear in the critical case N = p, which is mathematically more interesting. In this first paper we treat only the cases N > p and N = p, the case N < p will be considered in a companion work.

show abstract

Section: Preliminariesmentioning

confidence: 99%

Asymptotic analysis of the p-Laplacian flow in an exterior domain

Iagar

Vázquez

2009

Ann. Inst. H. Poincaré Anal. Non Linéaire

Self Cite

View full text Add to dashboard Cite

show abstract

“…We make the convention, also introduced in [12], that the dimensions and variables concerning the FPLE are named similarly as the correspondent ones for the VFDE, but with an overline. Since we deal only with radial solutions and variables, we will also accept noninteger dimensions n andn.…”

mentioning

confidence: 99%

“…Since we deal only with radial solutions and variables, we will also accept noninteger dimensions n andn. With these notations in mind, our starting point is the correspondence relations for radial solutions of the PME and PLE obtained in [12] and that we briefly recall here: (1.5)…”

mentioning

confidence: 99%

“…(ii) In [12], the values of D 1 and D 2 are different since there we start from the FPLE written in the usual form, without dividing by p − 1. We will keep this change in all the present paper and this will induce small differences in some expressions with respect to [12].…”

mentioning

confidence: 99%

“…We will keep this change in all the present paper and this will induce small differences in some expressions with respect to [12].…”

mentioning

confidence: 99%

See 2 more Smart Citations

Radial equivalence and study of self-similarity for two very fast diffusion equations

Iagar

Sánchez

2009

Journal of Mathematical Analysis and Applications

Self Cite

View full text Add to dashboard Cite

In this paper we continue the study of the radial equivalence between the porous medium equation and the evolution p-Laplacian equation, begun in a previous work. We treat the cases m < 0 and p < 1. We perform an exhaustive study of self-similar solutions for both equations, based on a phase-plane analysis and the correspondences we discover.We also obtain special correspondence relations and self-maps for the limit case m = −1, p = 0, which is particularly important in applications in image processing. We also find self-similar solutions for the very fast p-Laplacian equation that have finite mass and, in particular, some of them that conserve mass, while this phenomenon is not true for the very fast diffusion equation.

show abstract

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

Vázquez

2017

Lecture Notes in Mathematics

Self Cite

102

View full text Add to dashboard Cite

We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel development, the theory of stochastic differential equations gives a foundation to the probabilistic study of diffusion.Nonlinear diffusion equations have played an important role not only in theory but also in physics and engineering, and we focus on a relevant aspect, the existence and propagation of free boundaries. Due to our research experience, we use the porous medium and fast diffusion equations as case examples.A large part of the paper is devoted to diffusion driven by fractional Laplacian operators and other nonlocal integro-differential operators representing nonlocal, long-range diffusion effects. Three main models are examined (one linear, two nonlinear), and we report on recent progress in which the author is involved.

show abstract

Radial equivalence for the two basic nonlinear degenerate diffusion equations

Cited by 33 publications

References 18 publications

Asymptotic analysis of the p-Laplacian flow in an exterior domain

Asymptotic analysis of the p-Laplacian flow in an exterior domain

Radial equivalence and study of self-similarity for two very fast diffusion equations

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

Contact Info

Product

Resources

About