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

From the introduction: "We continue here the investigation initiated earlier by us [Trans. Amer. Math. Soc. 314 (1989), no. 1, 187–224] concerning the solvability of the Cauchy problem and the existence of initial traces for nonnegative weak solutions of the nonlinear evolution equation (1) ut − div(|Du| p−2 Du)=0 in ST ≡ RN ×(0,T), 0 2, both in terms of results and techniques. The main difference stems from the fact that, unlike solutions in the case p>2, solutions of (1) are not, in general, locally bounded.''

Non-negative solutions of the evolution p-Laplacian equation. Initial traces and cauchy problem when 1<p<2

On considere le probleme de Cauchy pour l'equation de diffusion non lineaire ∂u/∂t−Δ(u|u| m−1 )=0 sur (0, ∞)×R N , u(0)=u 0 quand 0<m<1. On montre qu'il existe une solution temporelle globale pour une fonction localement integrable u 0 . On montre l'unicite dans cette classe

The Cauchy problem for ut=Δum when 0<m<1

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

M. Pierre

Author Tools

Chat about Author

Papers

The Cauchy problem for ut=Δum when 0<m<1