73 research outputs found
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
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