Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D

Abstract

We consider nonlinear parabolic equations involving fractional diffusion of the form $${\partial_t u + {(-\Delta)}^{s} \Phi(u)= 0,}$$ with $${0 < s < 1}$$ , and solve an open problem concerning the existence of solutions for very singular nonlinearities $${\Phi}$$ in power form, precisely $${\Phi'(u)=c\,u^{-(n+1)}}$$ for some $${0 < n < 1}$$ . We also include the logarithmic diffusion equation $${\partial_t u + {(-\Delta)}^{s} \log(u)= 0}$$ , which appears as the case $${n=0}$$ . We consider the Cauchy problem with nonnegative and integrable data $${u_0(x)}$$ in one space dimension, since the same problem in higher dimensions admits no nontrivial solutions according to recent results of the author and collaborators. The limit solutions we construct are unique, conserve mass, and are in fact maximal solutions of the problem. We also construct self-similar solutions of Barenblatt type, which are used as a cornerstone in the existence theory, and we prove that they are asymptotic attractors (as $${t\to\infty}$$ ) of the solutions with general integrable data. A new comparison principle is introduced.