Abstract
In this paper, we study the partial differential equation
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.
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Ruvalcaba-Robles, E., Villa-Morales, J. Positive solutions and persistence of mass for a nonautonomous equation with fractional diffusion. Arab. J. Math. 6, 109–122 (2017). https://doi.org/10.1007/s40065-017-0167-3
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DOI: https://doi.org/10.1007/s40065-017-0167-3