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 $$\phi $$ .