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Applied Mathematics and Mechanics(English Edition) 2007年第1期28卷 111-118页

作者：石佩虎王明新 Department of Mathematics Southeast University Nanjing 210096 P. R. China

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation （ODE）. Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.

关键词： fast diffusion equation gradient absorption self-similar singular solution,very singular solution

The Second Critical Exponent for a fast diffusion equation with Potential

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Journal of Mathematical Research with Applications 2012年第6期32卷 715-722页

作者： Chunxiao YANG1,2, Jin’ge YANG1, Sining ZHENG1, 1. School of Mathematical Science, Dalian University of Technology, Liaoning 116024, P. R. China 2. Department of Mathematics, Xi’an University of Architecture and Technology,Shaanxi 710055, P. R. China School of Mathematical Science Dalian University of Technology Liaoning 116024

This paper considers a fast diffusion equation with potential ut= um V (x)um+upin Rn×(0,T), where 1 2αm+n 1, n ≥ 2, V (x) ～ω|x|2with ω≥ 0 as |x| →∞,and α is the positive root of αm(αm + n 2) ω = 0. The critical Fujita exponent was determined as pc= m +2αm+nin a previous paper of the authors. In the present paper,we establish the second critical exponent to identify the global and non-global solutions in their co-existence parameter region p > pcvia the critical decay rates of the initial *** u0(x) ～ |x| aas |x| →∞, it is shown that the second critical exponent a =2p m,independent of the potential parameter ω, is quite different from the situation for the critical exponent pc.

关键词： the second critical exponent fast diffusion equation potential global solutions blow-up.

Localization for a fast diffusion equation with Absorption

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Communications in Mathematical Research 1999年第2期 27-32页

作者： Yuan Hongjun (Institute of Mathematics, Jilin University, Changchun, 130012)

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒｗｅｃｏｎｓｉｄｅｒｔｈｅｆａｓｔｄｉｆｕｓｉｏｎｅｑｕａｔｉｏｎｗｉｔｈａｂｓｏｒｐｔｉｏｎｕｔ＝Δｕｍ－ｕｐ（１）ｉｎＱ＝Ｒｎ×（０，＋∞）ｗｉｔｈｔｈｅｉｎｉ... 详细信息

§１．ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｉｓｐａｐｅｒｗｅｃｏｎｓｉｄｅｒｔｈｅｆａｓｔｄｉｆｕｓｉｏｎｅｑｕａｔｉｏｎｗｉｔｈａｂｓｏｒｐｔｉｏｎｕｔ＝Δｕｍ－ｕｐ（１）ｉｎＱ＝Ｒｎ×（０，＋∞）ｗｉｔｈｔｈｅｉｎｉｔｉａｌｃｏｎｄｉｔｉｏｎｕ（ｘ，...

关键词： Localization Nonlinear diffusion equation fast diffusion equation

Critical Exponents for fast diffusion equations with Nonlinear Boundary Sources

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Communications in Mathematical Research 2011年第2期27卷 97-104页

作者： WANG LU-SHENG WANG ZE-JIA School of Mathematics Jilin University Changehun 130012

In this paper,we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit *** are interested in the critical global exponent q_o and the critical Fujita exponent q_c for the problem considered,and show that q_o=q_c for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources,which is quite different from the known results that q_o〈q_c for the onedimensional case;moreover,the value is different from the slow case.

关键词： exterior domain critical global exponent critical Fujita exponent fast diffusion equation

The Moser-Trudinger-Onofri Inequality

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Chinese Annals of Mathematics,Series B 2015年第5期36卷 777-802页

作者： Jean DOLBEAULT Maria J.ESTEBAN Gaspard JANKOWIAK Ceremade CNRS UMR 7534 and Université Paris-Dauphine Place de Lattre de Tassigny

This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary *** new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods(in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev *** the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally,a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.

关键词： Moser-Trudinger-Onofri inequality Duality Mass transportation fast diffusion equation Rigidity

Harnack Differential Inequalities for the Parabolic equation u_t= LF(u) on Riemannian Manifolds and Applications

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Acta Mathematica Sinica,English Series 2017年第5期33卷 620-634页

作者： wen wang school of mathematics and statistics hefei normal universityHefei 230601P.R.China school of mathematical science university of science and technology of chinaHefei 230026P.R.China

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F>0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.

关键词： Parabolic equation Li–Yau type Harnack differential inequality local Hamilton type gradient estimate fast diffusion equation Porous media equation