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

In this paper, we study the partial differential equation 1 $$\begin{aligned} \begin{aligned} \partial _tu&= k(t)\Delta _\alpha u - h(t)\varphi (u),\\ u(0)&= u_0. \end{aligned} \end{aligned}$$ 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.