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)Δ_αu - h(t)φ(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 $\phi :\mathbb{R}\to [0,\infty )$ 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 $u_0\ge 0$. 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 $\phi$ is a power function.

18 pages