Arabian Journal of Mathematics

, Volume 6, Issue 2, pp 109–122 | Cite as

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

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Abstract

In this paper, we study the partial differential equation
$$\begin{aligned} \begin{aligned} \partial _tu&= k(t)\Delta _\alpha u - h(t)\varphi (u),\\ u(0)&= u_0. \end{aligned} \end{aligned}$$
(1)
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 \(0\le u_0\in C_b(\mathbb {R}^d)\cap L^1(\mathbb {R}^d)\) 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 \(\lim _{t\rightarrow \infty }M(t)>0\), 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.

Mathematics Subject Classification

35K55 35K45 35B40 35K20 

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© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Departamento de Matemáticas y Física AguascalientesUniversidad Autónoma de AguascalientesAguascalientesMexico

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