Grzegorz Karch

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

Asymptotic Behaviour of Solutions to some Pseudoparabolic Equations

The aim of this paper is to investigate the behaviour as t→∞ of solutions to the Cauchy problem u t - Δu t - νΔu - (b,⊇u) =⊇.F(u), u(x, 0) = u o (x), where ν > 0 is a fixed constant, t ≥ 0, x ∈ R n . First, we prove that if u is the solution to the linearized equation, i.e. with ⊇.F (u) ≡ 0, then u decays like a solution for the analogous problem to the heat equation. Moreover, the long-time behaviour of u is described by the heat kernel. Next, analogous results are established for the non-linear equation with some assumptions imposed on F, p, and the initial condition u o .

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.

ON CONVERGENCE OF SOLUTIONS OF FRACTAL BURGERS EQUATION TOWARD RAREFACTION WAVES

In this paper, the large time behavior of solutions of the Cauchy problem for the one-dimensional fractal Burgers equation

Asymptotic Properties of Entropy Solutions to Fractal Burgers Equation

u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0

The

with

8\pi

\alpha\in (1,2)

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

is studied. It is shown that if the nondecreasing initial datum approaches the constant states

Large-time behaviour of solutions to non-linear wave equations: higher-order asymptotics

u_\pm

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

(