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. \u

Instantaneous shrinking in nonlinear diffusion-convection

The Cauchy problem for a nonlinear diffusion-convection equation is studied. The equation may be classified as being of degenerate parabolic type with one spatial derivative and a time derivative. It is shown that under certain conditions solutions of the initial-value problem exhibit instantaneous shrinking. This is to say, at any positive time the spatial support of the solution is bounded above, although the support of the initial data function is not. This is a phenomenon which is normally only associated with nonlinear diffusion with strong absorption. In conjunction, a previously unreported phenomenon is revealed. It is shown that for a certain class of initial data functions there is a critical positive time such that the support of the solution is unbounded above at any earlier time, whilst the opposite is the case at any later time

Instantaneous shrinking and single point extinction for viscous Hamilton-Jacobi equations with fast diffusion

International audienceFor a large class of non-negative initial data, the solutions to the quasilinear viscous Hamilton-Jacobi equation $\partial_t u-\Delta_p u+|\nabla u|^q=0$ in $(0,\infty)\times\mathbb{R}^N$ are known to vanish identically after a finite time when $2N/(N+1) 0$, the positivity set of $u(t)$ is a bounded subset of $\mathbb{R}^N$ even if $u_0 > 0$ in $\mathbb{R}^N$. This decay condition on $u_0$ is also shown to be optimal by proving that the positivity set of any solution emanating from a positive initial condition decaying at a slower rate as $|x|\to\infty$ is the whole $\mathbb{R}^N$ for all times. The time evolution of the positivity set is also studied: on the one hand, it is included in a fixed ball for all times if it is initially bounded (\emph{localization}). On the other hand, it converges to a single point at the extinction time for a class of radially symmetric initial data, a phenomenon referred to as \emph{single point extinction}. This behavior is in sharp contrast with what happens when $q$ ranges in $[p-1,p/2)$ and $p\in (2N/(N+1),2]$ for which we show \emph{complete extinction}. Instantaneous shrinking and single point extinction take place in particular for the semilinear viscous Hamilton-Jacobi equation when $p=2$ and $q\in (0,1)$ and seem to have remained unnoticed

Instantaneous shrinking of the support of energy solutions

AbstractThe Cauchy problem for a class of nonlinear diffusion-reaction equations is studied. The equations may be classified as being of degenerate parabolic type. It is shown that under certain conditions solutions of the problem exhibit instantaneous shrinking. This is to say, at any positive time the spatial support of the solution is bounded above, although the support of the initial data function is not. We also provide some estimates of the behavior of the free boundary

The reversing of interfaces in slow diffusion processes with strong absorbtion

This paper considers a family of one-dimensional nonlinear diffusion equations with absorption. In particular, the solutions that have interfaces that change their direction of propagation are examined. Although this phenomenon of reversing interfaces has been seen numerically, and some special exact solutions have been obtained, there was previously no analytical insight into how this occurs in the general case. The approach taken here is to seek self-similar solutions local to the interface and local to the reversing time. The analysis is split into two parts, one for the solution prior to the reversing time and the other for the solution after the reversing time. In each case the governing PDE is reduced to an ODE by introducing a self-similar coordinate system. These ODEs do not readily admit any nontrivial exact solutions and so the asymptotic behavior of solutions is studied. By doing this the adjustable parameters, or degrees of freedom, which may be used in a numerical shooting scheme are determined. A numerical algorithm is then proposed to furnish solutions to the ODEs and hence the PDE in the limit of interest. As examples of physical problems in which a PDE of this type may be used as a model the authors study the spreading of a viscous film under gravity and subject to evaporation, the dispersion of a population, and a nonlinear heat conduction problem. The numerical algorithm is demonstrated using these examples. Results are also given on the possible existence of self-similar solutions and types of reversing behavior that can be exhibited by PDEs in the family of interest