Extinction and decay estimates for viscous Hamilton-Jacobi equations in \({\mathbb{R}}^N\) (Q2781258)

The Cauchy problem NEWLINE\[NEWLINE\begin{aligned} u_t-\Delta u+|\nabla u|^p&=0\quad \text{in }(0,+\infty)\times\mathbb{R}^N,\tag{1} \\ u(0)&=u_0\quad \text{in }\mathbb{R}^N,\tag{2}\end{aligned}NEWLINE\]NEWLINE where \(p\in (0,+\infty)\) and \(u_0\) is a nonnegative function in \({\mathcal BC}(\mathbb{R}^n)\cap L^1(\mathbb{R}^N)\), is considered. Here, \({\mathcal BC}(\mathbb{R}^N)\) denotes the space of bounded and continuous functions in \(\mathbb{R}^N\). For such initial data, the existence and the uniqueness of nonnegative classical solutions to (1)-(2) have been obtained by Gilding, Guedda and Kersner. Within the framework of nonnegative solutions the term \(|\nabla u|^p\) in (1) acts as an absorption term, and the smaller the exponent \(p\) is, the stronger is the absorption. The aim of this work is to investigate some qualitative properties of nonnegative solutions to (1)-(2) according to the values of \(p\). More precisely, the authors proved that any nonnegative solution to (1)-(2) with initial data in \({\mathcal BC}(\mathbb{R}^N)\cap L^1(\mathbb{R}^N)\) vanishes identically after a finite time when \(p\in (0,N/N(N+1))\), this property called extinction in finite time.