Rate of Convergence to Barenblatt Profiles for the Fast Diffusion Equation with a Critical Exponent

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non--linear Fokker--Planck equation, we find a continuum of algebraic rates of convergence to a self--similar profile. These rates depend explicitly on the spatial decay rates of initial data. This improves a previous result on slow convergence for the critical fast diffusion equation ({\sc Bonforte et al}. in Arch Rat Mech Anal 196:631--680, 2010) and provides answers to some open problems

Backward selfsimilar solutions of supercritical parabolic equations

AbstractWe consider the exponential reaction–diffusion equation in space-dimension n∈(2,10). We show that for any integer k≥2 there is a backward selfsimilar solution which crosses the singular steady state k-times. The same holds for the power nonlinearity if the exponent is supercritical in the Sobolev sense and subcritical in the Joseph–Lundgren sense

Fast diffusion equation: uniqueness of solutions with a moving singularity

We focus on open questions regarding the uniqueness of distributional solutions of the fast diffusion equation (FDE) with a given source term. When the source is sufficiently smooth, the uniqueness follows from standard results. Assuming that the source term is a measure, the existence of different classes of solutions is known, but in many cases, their uniqueness is an open problem. In our work, we focus on the supercritical FDE and prove the uniqueness of distributional solutions with a Dirac source term that moves along a prescribed curve. Moreover, we extend a uniqueness results for the subcritical FDE from standing to moving singularities.Comment: 11 page