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

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.