Transformations of self-similar solutions for porous medium equations of fractional type

Abstract

We consider four different models of nonlinear diffusion equations involving fractional Laplacians and study the existence and properties of classes of self-similar solutions. Such solutions are an important tool in developing the general theory. We introduce a number of transformations that allow us to map complete classes of solutions of one equation into those of another one, thus providing us with a number of new solutions, as well as interesting connections. Special attention is paid to the property of finite propagation.

Porous medium equation with nonlocal pressure in a bounded domain

“…The same equation as in [18] appears in a one-dimensional model in dislocation theory that has also been studied by Biler et al [6]. Later mathematical works include [19,17,20], where regularity and asymptotic behaviour are established, paper [5] that treats the case m 1 = 1, m 2 > max{ 1−2s 1−s , 2s−1 N }, and the works [36,38,39] that treat the cases where m 1 = 1, and [37] that treats general exponents, see also [27].…”

“…• Question of finite speed of propagation, cf. works [37,38,39] for problems posed in the whole space. Regularity of free boundary problems, with open questions even for PME with nonlocal pressure, [18].…”

Porous Medium Equation with Nonlocal Pressure

“…In a previous work [55] we have established three main types of self-similar solutions for model (M1) depending on the range of the parameter m, but always restricted to the range m < 2. The first type are functions that are positive for all times, while the second type are functions that extinguish in finite time, both separated by a transition type.…”

“…These ones can be transformed in a similar manner to corresponding self-similar solutions of the model (M1). Rigorous proofs with complete computations can be found in [55]. Self-similar solutions do not have an explicit formula, except very particular cases of exponents m = m(s) explicitly computed by Huang in [37].…”

Initial-boundary value problem for a fractional type degenerate heat equation

“…A different approach to prove existence based on gradient flows has been developed by Lisini, Mainini and Segatti (see [20]). Then the model has been generalized in [26] [27] [28] [29] [30]. Uniqueness is still open in general, but under some truly restrictive regularity assumption is proven in [31].…”