Non-existence and instantaneous extinction of solutions for singular nonlinear fractional diffusion equations

Abstract

We show non-existence of solutions of the Cauchy problem in $\mathbb {R}^N$ for the nonlinear parabolic equation involving fractional diffusion $\partial _t u + {{\mathrm{(-\Delta )}}}^{s}\phi (u)= 0,$ with $0<s<1$ and very singular nonlinearities $\phi$. It is natural to consider nonnegative data and solutions. More precisely, we prove that when $\phi (u)=-1/u^n$ with $n>0$, or $\phi (u) = \log u$, and we take nonnegative $L^1$ initial data, there is no solution of the problem in any dimension $N\ge 2$. In one space dimension the situation is not so radical, and we find the optimal range of non-existence when $N=1$ in terms of s and n. As a complement, non-existence is then proved for more general nonlinearities $\phi$, and it is also extended to the related elliptic problem of nonlinear nonlocal type: $u + {{\mathrm{(-\Delta )}}}^{s}\phi (u) = f$ with the same type of nonlinearity $\phi$.