The porous medium equation

CONTENTSIntroduction Chapter I. Existence and uniqueness of generalized solutions of the Cauchy problem and of initial boundary-value problems ??1.1. Theorems of existence and non-existence ??1.2. Uniqueness theorems Chapter II. Estimates and differential properties of generalized solutions ??2.1. The character of the dependence on the data ??2.2. Regularity and non-regularity Chapter III. Behaviour of generalized solutions for small values of time ??3.1. Finite rate of propagation of perturbations ??3.2. Inertia ??3.3. Shrinking of the size of the support Chapter IV. Behaviour of generalized solutions for large values of time ??4.1. Stabilization in finite time ??4.2. Localization of perturbations ??4.3. Asymptotic formulae References

http://iopscience.iop.org/article/10.1070/RM1987v042n02ABEH001309/pdf

Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations

We consider non-negative solutions of the fast diffusion equation ut = Δum with m ∈ (0, 1) in the Euclidean space \({{\mathbb R}^d}\), d ≧ 3, and study the asymptotic behavior of a natural class of solutions in the limit corresponding to t → ∞ for m ≧ mc = (d − 2)/d, or as t approaches the extinction time when m < mc. For a class of initial data, we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if m ≧ mc, or close enough to the extinction time if m < mc. Such results are new in the range m ≦ mc where previous approaches fail. In the range mc < m < 1, we improve on known results.

/pdf/asymptotics-of-the-fast-diffusion-equation-via-entropy-2uus11esmn.pdf

Asymptotics of the Fast Diffusion Equation via Entropy Estimates

The classification of locally conformally flat Yamabe solitons

We give a simple proof of the λ = d - 2 cases of the sharp Hardy-Littlewood-Sobolev inequality for d≥3, and the sharp Logarithmic Hardy-Littlewood-Sobolev inequality for d = 2 via a monotone flow governed by the fast diffusion equation.

/pdf/hardy-littlewood-sobolev-inequalities-via-fast-diffusion-zutl99ufy5.pdf

Hardy-Littlewood-Sobolev inequalities via fast diffusion flows

/pdf/the-cauchy-problem-for-d-when-0-1-18crlcmqh0.pdf

The Cauchy problem for _{}=Δ^{} when 0<<1

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The Cauchy problem for _{}=Δ^{} when 0<<1