Robert Kersner

Travelling waves in nonlinear diffusion-convection reaction

The study of travelling waves or fronts has become an essential part of the mathematical analysis of nonlinear diffusion-convection-reaction processes. Whether or not a nonlinear second-order scalar reaction-convection-diffusion equation admits a travelling-wave solution can be determined by the study of a singular nonlinear integral equation. This article is devoted to demonstrating how this correspondence unifies and generalizes previous results on the occurrence of travelling-wave solutions of such partial differential equations. The detailed comparison with earlier results simultaneously provides a survey of the topic. It covers travelling-wave solutions of generalizations of the Fisher, Newell-Whitehead, Zeldovich, KPP and Nagumo equations, the Burgers and nonlinear Fokker-Planck equations, and extensions of the porous media equation.

On a nonlinear degenerate parabolic equation in infiltration or evaporation through a porous medium

The main result of the paper is the uniqueness of nonnegative solutions of the Cauchy problem and of the first and mixed boundary value problems for a class of degenerate parabolic equations which includes the model equation ut=(um)xx+(un)x, where m≥1 and n>0. In particular, n is allowed to be smaller than one. The proof is based on a refined test function argument. The condition that u be nonnegative is crucial, but the restriction to one space variable is not.

The Cauchy problem for ut=Δu+|∇u|q

With q a positive real number, the nonlinear partial differential equation in the title of the paper arises in the study of the growth of surfaces. In that context it is known as the generalized deterministic KPZ equation. The paper is concerned with the initial-value problem for the equation under the assumption that the initial-data function is bounded and continuous. Results on the existence, uniqueness, and regularity of solutions are obtained.

Local and Global Solvability of a Class of Semilinear Parabolic Equations

PIERRE BARAS Laborutoire IMAG, Tour des Mathkmatiques, B. P. 68. 38402, Saint-Martin D’Heres, Cedex, France ROBERT KERSNER Computer and Automation Institute, Hungarian Academy of Sciences, P. 0. Box 63. H-1502, Budapest, Hungary Received February 11, 1986; revised September 5, 1986

The nonlinear heat equation with absorption: Effects of variable coefficients

Abstract We consider the nonnegative solutions to the nonlinear degenerate parabolic equation ut = (D(x, t)um − 1ux)x − b(x, t)up with m > 1, 0

On the behaviour and cases of nonexistence of the free boundary in a semibounded porous medium

The authors consider the Fokker-Planck equation ut=(um)xx+b(uλ)x, x>0, t>0, with initial and boundary data u(x,0)=u0(x), x>0, u(0,t)=u1(t), t>0, u0 having its support in a bounded interval. They concentrate on the case 0 0 it is shown that if u1 tends to zero as t→∞, then the free boundary tends to zero. If u1 vanishes in a finite time, so does the free boundary. The possibility that the free boundary tends to infinity is also discussed. Moreover, conditions are found on m,λ and on u1 such that the free boundary can be estimated from above (localization) and from below by a positive constant. When b 1 the free boundary is known to start from the right endpoint of suppu0).

Instantaneous extinction, step discontinuities and blow-up

This note concerns reaction–diffusion processes which display remarkable behaviour. Everywhere the concentration, density or temperature exceeds some critical level until at some moment in time it decreases to the critical level at one point in space. At this instant, the complete profile immediately drops to the critical level at every point in space, and then remains there.

Wavefronts and unbounded waves for power-law equations

With the porous media equation as prototype, equations of the class (1.1) with power-law coefficients have attracted much interest to date. In this chapter, we shall classify all the global monotonic travelling-wave solutions decreasing to 0 and all the unbounded monotonic semi-wavefront solutions decreasing to 0 for equations of this type. We begin with the power-law convection-diffusion equation, proceed to the power-law reaction-diffusion equation with linear convection, and, end with the full equation.

Power-law equations

We turn now to the application of the integral equation (1.9) for the definitive analysis of semi-wavefront solutions for two specific classes of equation (1.1). The first of these is