Global existence and persistence of mass for a nonlinear equation with fractional Laplacian

TL;DR

This paper establishes conditions for the global existence, persistence of mass, and non-vanishing of solutions for a nonlinear fractional Laplacian PDE, extending understanding of long-term behavior of such equations.

Contribution

It provides new results on global solutions and mass persistence for a fractional Laplacian PDE with nonlinearities, including conditions ensuring non-vanishing mass over time.

Findings

Global classical solutions exist for non-negative initial data.

Mass of solutions remains positive and does not vanish in finite time.

Conditions under which the mass persists indefinitely are identified.

Abstract

In this paper we study the partial differential equation \begin{equation} \begin{split} \partial_tu &= k(t)\Delta_\alpha u - h(t)\varphi(u), u(0) &= u_0. \end{split} \end{equation} Here $\Delta_\alpha$ is the fractional Laplacian, $k,h:[0,\infty)\to[0,\infty)$ are continuous functions and $\varphi:\mathbb{R}\to[0,\infty)$ is a convex differentiable function. If $u_0\in C_b(\mathbb{R}^d)\cap L^1(\mathbb{R}^d)$ we prove that the above equation has a classical global solution and is non-negative if $u_0\geq0$. Imposing some restrictions on the parameters we prove that the mass $M(t)=\int_{\mathbb{R}^d}u(t,x)dx$, $t>0$, of the system $u$ does not vanish in finite time, moreover we see that $\lim_{t\to\infty}M(t)>0$, under the restriction $\int_0^\infty h(s)ds<\infty$. A comparison result is also obtained for non-negative solutions, and as an application we get a better condition when $\varphi$ is a power function.