Publication Date:
2017-04-14
Description:
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.
Print ISSN:
2193-5343
Electronic ISSN:
2193-5351
Topics:
Mathematics
