Piotr Biler

Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws

Nonlocal conservation laws of the form ut + Lu +∇ · f( u)= 0, where −L is the generator of a Levy semigroup on L 1 (R n ), are encountered in continuum mechanics as model equations with anomalous diffusion. They are generalizations of the classical Burgers equation. We study the critical case when the diffusion and nonlinear terms are balanced, e.g. L ∼ (−�) α/2 ,1 <α< 2, f( s)∼ s|s| r−1 , r = 1 + (α − 1)/n. The results include decay rates of solutions and their genuinely nonlinear asymptotic behavior as time t tends to infinity, determined by self-similar source solutions.  2001 Editions scientifiques et medicales Elsevier

Remark on the decay for damped string and beam equations

where A is an operator in a Hilbert space H, M is a real function. Our results improve those obtained by de Brito in a recent paper [l]. We briefly recall the notation and the assumptions in [l] referring to this paper for other information on physical meaning, existence and regularity theory, applications and bibliography. Let A be a self-adjoint positive operator in a (real) Hilbert space H with dense domain V. We assume that 5 > 0 is the least eigenvalue of A so a(u) = (Au, u) 3 51~1~ for u E V. Let M E Cl@+) be a real nondecreasing function such that M(s) 2 p + ks, k z 0, p-real. We have, with the notation

Large mass self-similar solutions of the parabolic–parabolic Keller–Segel model of chemotaxis

In two space dimensions, the parabolic–parabolic Keller–Segel system shares many properties with the parabolic–elliptic Keller–Segel system. In particular, solutions globally exist in both cases as long as their mass is less than a critical threshold Mc. However, this threshold is not as clear in the parabolic–parabolic case as it is in the parabolic–elliptic case, in which solutions with mass above Mc always blow up. Here we study forward self-similar solutions of the parabolic–parabolic Keller–Segel system and prove that, in some cases, such solutions globally exist even if their total mass is above Mc, which is forbidden in the parabolic–elliptic case.

Existence and asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux boundary conditions

U(X, 0) = U&C) 2 0 (3) from which (D(x, 0) is determined via (la) and (2b). The systems of this type appear in the theory of semiconductors: [l-4] or in electrochemistry (where U, or rather ut, u2, . . . for more general systems involving several unknown functions U, are the densities of charge carriers: electrons, holes, ions), and in groundwater flow problems: [5, 61 (where u is the density of a fluid in a porous medium). The parabolic equation (la) is flux conservative and the boundary condition (2a) is the physically relevant no-flux condition, hence the total charge or mass (according to the physical interpretation)

Nonlinear Diffusion of Dislocation Density and Self-Similar Solutions

We study a nonlinear pseudodifferential equation describing the dynamics of dislocations in crystals. The long time asymptotics of solutions is described by the self-similar profiles.

The

We study the properties and the large time asymptotics of radially symmetric solutions of a chemotaxis system in a disc of R 2 when the parameter is either critical and equal to 8… or subcritical.

8\pi

We study properties of solutions of the system ut =r (#ru +ur’); ’ = u; E = M# + 1 Z u’dx: This system was proposed by Chavanis, Sommeria and Robert for description of evolution of density of a system of gravitating particles. In physical interpretation u(x;t), ’(x;t) are the density and the gravitational potential, respectively. The temperature #(t) is uniform in the domain , where the problem is considered. M is the total mass and E is the energy of the particles. We are interested in the existence of global solutions, blow-up phenomena and stationary solutions of our system. The results have been obtained jointly with Piotr Biler (Uniwersytet Wroc lawski) and Ignacio Guerra (Universidad de Chile).

-problem for radially symmetric solutions of a chemotaxis model in a disc

A degenerate nonlinear nonlocal evolution equation is considered; it can be understood as a porous medium equation whose pressure law is nonlinear and nonlocal. We show the existence of sign-changing weak solutions to the corresponding Cauchy problem. Moreover, we construct explicit compactly supported self-similar solutions which generalize Barenblatt profiles—the well-known solutions of the classical porous medium equation.

Global and exploding solutions in a model of self-gravitating systems

We review Streater’s energy-transport models which describe the temporal evolution of the density and temperature of a cloud of gravitating particles, coupled to a mean field Poisson equation. In particular we consider the existence of stationary solutions in a bounded domain with given energy and mass. We discuss the influence of the dimension and geometry of the domain on existence results.

The Nonlocal Porous Medium Equation: Barenblatt Profiles and Other Weak Solutions

We propose some nonlinear parabolic-elliptic systems modelling the evolution of the density of particles (charged or massive) interacting with themselves, and coupled to a temperature field. These models are thermodynamically consistent, i.e., they obey the first and the second laws of thermodynamics. We study their steady states and the asymptotic behaviour for large time.