Initial trace of solutions of Hamilton-Jacobi parabolic equation with absorption

TL;DR

This paper investigates the initial trace problem for nonnegative solutions of a Hamilton-Jacobi parabolic equation with absorption, establishing measure-theoretic properties, existence of extremal solutions, and growth behaviors near initial time.

Contribution

It characterizes the initial trace as a Radon measure for certain q, constructs extremal solutions for a range of measures, and describes growth rates of solutions near initial time.

Findings

Trace is a Radon measure when q ≤ 1.

Existence of minimal and maximal solutions for certain measures.

Solutions exhibit specific growth rates near t=0.

Abstract

Here we study the initial trace problem for the nonnegative solutions of the equation \[ u\_{t}-\Delta u+|\nabla u|^{q}=0 \] in $Q\_{\Omega,T}=\Omega\times\left( 0,T\right) ,$ $T\leqq\infty,$ where $q>0,$ and $\Omega=\mathbb{R}^{N},$ or $\Omega$ is a smooth bounded domain of $\mathbb{R}^{N}$ and $u=0$ on $\partial\Omega\times\left( 0,T\right) .$ We can define the trace at $t=0$ as a nonnegative Borel measure $(\mathcal{S} ,u\_{0}),$ where $S$ is the closed set where it is infinite, and $u\_{0}$ is a Radon measure on $\Omega\backslash\mathcal{S}.$ We show that the trace is a Radon measure when $q\leqq1.$ For $q\in(1,(N+2)/(N+1)$ and any given Borel measure, we show the existence of a minimal solution, and a maximal one on conditions on $u\_{0}.$ When $\mathcal{S}$ $=\overline{\omega}\cap\Omega$ and $\omega$ is an open subset of $\Omega,$ the existence extends to any $q\leqq2$ when $u\_{0}\in L\_{loc}^{1}(\Omega)$ and any $q>1$ when $u\_{0}=0$. In particular there exists a self-similar nonradial solution with trace $(\mathbb{R}^{N+},0),$ with a growth rate of order $\left\vert x\right\vert ^{q^{\prime}}$ as $\left\vert x\right\vert \rightarrow\infty$ for fixed $t.$ Moreover we show that the solutions with trace $(\overline{\omega},0)$ in $Q\_{\mathbb{R}^{N},T}$ may present near $t=0$ a growth rate of order $t^{-1/(q-1)}$ in $\omega$ and of order $t^{-(2-q)/(q-1)}$ on $\partial \omega.$