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 $\partial_t u + (-\Delta)^s \phi(u)= 0,$ with $0<s<1$ and very singular nonlinearities $\phi$ . 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 (nonnegative) solution of the problem in any dimension $N\ge 2$. We find the range of non-existence when $N=1$ in terms of $s$ and $n$. The range of exponents that we find for non-existence both for parabolic and elliptic equations are optimal. Non-existence is then proved for more general nonlinearities $\phi$, and it is also extended to the related elliptic problem of nonlinear nonlocal type: $u + (-\Delta)^s \phi(u) = f$ with the same type of nonlinearity $\phi$.