Positive solutions and persistence of mass for a nonautonomous equation with fractional diffusion

Abstract

In this paper, we study the partial differential equation 1∂tu=k(t)Δαu-h(t)φ(u),u(0)=u0.\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\begin{aligned} \\partial _tu&= k(t)\\Delta _\\alpha u - h(t)\\varphi (u),\\\\ u(0)&= u_0. \\end{aligned} \\end{aligned}$$\\end{document}Here Delta _alpha =-(-Delta )^{alpha /2}, 0<alpha <2, is the fractional Laplacian, k,h:[0,infty )rightarrow [0,infty ) are continuous functions and varphi :mathbb {R}rightarrow [0,infty ) is a convex differentiable function. If 0le u_0in C_b(mathbb {R}^d)cap L^1(mathbb {R}^d) we prove that (1) has a non-negative classical global solution. Imposing some restrictions on the parameters we prove that the mass M(t)=int _{mathbb {R}^d}u(t,x)mathrm{d}x, t>0, of the system u does not vanish in finite time, moreover we see that lim _{trightarrow infty }M(t)>0, under the restriction int _0^infty h(s)mathrm{d}s<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.