Moments determine long-time asymptotics for diffusion equations

Abstract

The purpose of this thesis is to investigate relation between moments of initial data and long-time asymptotics of diffusion equations. More precisely, $L^1$-intermediate asymptotics for nonlinear diffusion equations, approximate solutions to the viscous Burgers equation and long-time asymptotics of the zero level set for the heat equation will be discussed. In Chapter 2, Newtonian potential is introduced in a relative sense for radial functions. This makes us possible to treat the potential theory for a larger class of functions in a unified manner for all dimensions $d\ge1$. For example, Newton`s theorem can be restated in a simpler form without concerning dimensions. The relative potential is then used to obtain the $L^1$-convergence order $O(t^{-1})$ as $t\to\infty$ for radially symmetric solutions to the porous medium and fast diffusion equations. Similar technique is also applied to radial solutions of the $p$-Laplacian equations to obtain the same convergence order. In Chapter 3, two kinds of approximate solutions to the heat equation are discussed. They will be used in the following chapters. In Chapter 4, relation between the moments and the asymptotic behavior of solutions to the viscous Burgers equation is investigated. The Burgers equation is a nonlinear problem having a special property; it can be transformed to a linear problem via the Cole-Hopf transformation. Our asymptotic analysis depends on the transformation. In the chapter an asymptotic approximate solution is constructed, which is given by the inverse Cole-Hopf transformation of a summation of $n$ heat kernels. The $k$ -th order moments of exact solution and the approximate solution are contracting with order $O\big((\sqrt{t})^{k-2n-1+1/p})$ in $L^p$- norm as $t\to\infty$. This asymptotics indicates that the convergence order is increased by a similarity scale whenever the order of controlled moments is increased by one. The theoretical asymptotic convergence orders are tested numerically....