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BLOW-UP OF THE SOLUTION FOR A CLASS OF porous medium equation WITH POSITIVE INITIAL ENERGY

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Acta Mathematica Scientia 2013年第4期33卷 1024-1030页

作者：吴秀兰高文杰 Institute of Mathematics Jilin University College of Mathematics Jilin Normal University

This paper deals with a class of porous medium equation ut=△u^m＋f（u）with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on... 详细信息

关键词： porous medium equation blow-up positive initial energy

COMPLEXITY OF ASYMPTOTIC BEHAVIOR OF SOLUTIONS FOR THE porous medium equation WITH ABSORPTION

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Acta Mathematica Scientia 2010年第6期30卷 1865-1880页

作者：尹景学王良伟黄锐 Department of Mathematics Jilin University School of Mathematical Sciences South China Normal University

In this paper we analyze the large time behavior of nonnegative solutions of the Cauchy problem of the porous medium equation with absorption ut - △um ＋ yup = 0,where γ≥0,m〉 1and P〉m＋2/N We will show that if γ=0 and 0〈μ〈 2N/n（m-1）＋2 or γ 〉 0 and 1/p-1 〈 μ 〈 2N/N（m-1）＋2 then for any nonnegative function φ in a nonnegative countable subset F of the Schwartz space S（RN）, there exists an initial-value u0 ∈ C（RN） with limx→∞ uo（x）= 0 such that φ is an w-limit point of the rescaled solutions tμ/2u（tβ, t）, Where β = 2-μ（m-1）/4.

关键词： complexity asymptotic behavior porous medium equation

Blow-up for a porous medium equation with Local Linear Boundary Dissipation

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Wuhan University Journal of Natural Sciences 2024年第2期29卷 95-105页

作者： YANG Jichen LIU Dengming School of Mathematics and Computational Science Hunan University of Science and TechnologyXiangtan 411201HunanChina

This article investigates the blow-up behaviors for a porous medium equation with a superlinear source and local linear boundary *** use of the concavity method,we establish sufficient conditions to guarantee the occurrence of the finite time blow-up ***,we show the existence of the finite time blow-up solutions for arbitrarily high initial ***,we derive the life span bounds(i.e.,the lower and upper bounds of the blow-up time).

关键词： porous medium equation blow-up behavior life span bounds

A Formulation of the porous medium equation with Time-Dependent Porosity: A Priori Estimates and Regularity Results

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Applied Mathematics 2024年第10期15卷 745-763页

作者： Koffi B. Fadimba Department of Computer Science Engineering and Mathematics University of South Carolina Aiken Aiken USA

We consider a generalized form of the porous medium equation where the porosity ϕis a function of time t: ϕ=ϕ(x,t): ∂(ϕS)∂t−∇⋅(k(S)∇S)=Q(S).In many works, the porosity ϕis either assumed to be independent of (or to depend very little of) the time variable t. In this work, we want to study the case where it does depend on t(and xas well). For this purpose, we make a change of unknown function V=ϕSin order to obtain a saturation-like (advection-diffusion) equation. A priori estimates and regularity results are established for the new equation based in part on what is known from the saturation equation, when ϕis independent of the time t. These results are then extended to the full saturation equation with time-dependent porosity ϕ=ϕ(x,t). In this analysis, we make explicitly the dependence of the various constants in the estimates on the porosity ϕby the introduced transport vector w, through the change of unknown function. Also we do not assume zero-flux boundary, but we carry the analysis for the case Q≡0.

关键词： porous medium equation Porosity Saturation equation A Priori Estimates Regularity Results

Fujita Exponent for porous medium equation with Convection and Nonlinear Boundary Condition

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Northeastern Mathematical Journal 2003年第4期19卷 387-395页

作者：王泽佳尹景学 Department of Mathematics Jilin University Changchun 130012 Department of Mathematics Jilin University Changchun 130012his paper is concerned with the critical exponent of the porous medium equation with convection and nonlinear boundary condition. It is shown that the coefficient of the lower order term is an important factor that determines the critical exponent.

This paper is concerned with the critical exponent of the porous medium equation with convection and nonlinear boundary condition. It is shown that the coefficient of the lower order term is an important factor that d... 详细信息

关键词： porous medium equation critical exponent blow-up

Blow-up and Global Solutions for a porous medium equation Under Robin Boundary Conditions

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Blow-up and Global Solutions for a Porous Medium Equation Un...

Local Aronson-Benilan Estimates for porous medium equations under Ricci Flow

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作者： Jinlan Hao Lingling Zhang Department of Mathematics Taiyuan University of Technology State Key Laboratory of Explosion Science and Technology Beijing Institute of Technology

This paper deals with blow-up and global solutions for a porous medium equation under Robin boundary *** constructing auxiliary functions and using modified differential inequality techniques,we establish conditions to ensure that the solution blows up in some finite time or remains ***,an upper and a lower bound for blow-up time are ***,an example is given to illustrate the applications of obtained results.

关键词： porous medium equation Blow-up Global solution

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Journal of Partial Differential equations 2011年第4期24卷 324-333页

作者： ZHU Xiaobao Institute of Mathematics Academy of Mathematics and Systems Sciences ChineseAcademy of Sciences Beijing 100190 China.

In this work we derive local gradient estimates of the Aronson-Benilan type for positive solutions of porous medium equations under Ricci flow with bounded Ricci curvature. As an application, we derive a Harnack type ... 详细信息

关键词： Aronson-Benilan estimate porous medium equation Ricci flow Harnack inequality.

Nonlinear Diffusion and Transient Osmosis

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Communications in Theoretical Physics 2011年第8期56卷 352-366页

作者： Akira Igarashi Lamberto Rondoni Antonio Botrugno Marco Pizzi Dipartimento di Matematica Politecnico di Torino INFN Sezione di Torino Kavli Institute for Theoretical Physics China CAS Eltek Group

We investigate both analytically and numerically the concentration dynamics of a solution in two containers connected by a narrow and short channel, in which diffusion obeys a porous medium equation. We also consider the variation of the pressure in the containers due to the flow of matter in the channel. In particular, we identify a phenomenon, which depends on the transport of matter across nano-porous membranes, which we call ＂transient osmosis＂. We find that nonlinear diffusion of the porous medium equation type allows numerous different osmotic-like phenomena, which are not present in the case of ordinary Fickian diffusion. Experimental results suggest one possible candidate for transiently osmotic processes.

关键词： anomalous transport porous medium equation osmotic pressure

porous medium Flow with Both a Fractional Potential Pressure and Fractional Time Derivative

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Chinese Annals of Mathematics,Series B 2017年第1期38卷 45-82页

作者： Mark ALLEN Luis CAFFARELLI Alexis VASSEUR Department of Mathematics The University of Texas at AustinAustinTX 78712USA

The authors study a porous medium equation with a right-hand side. The operator has nonlocal diffusion effects given by an inverse fractional Laplacian *** derivative in time is also fractional and is of Caputo-type, which takes into account"memory". The precise model isD_t~αu- div(u(-Δ)^(-σ)u) = f, 0 < σ <1/*** paper poses the problem over {t ∈ R^+, x ∈ R^n} with nonnegative initial data u(0, x) ≥0 as well as the right-hand side f ≥ 0. The existence for weak solutions when f, u(0, x)have exponential decay at infinity is proved. The main result is H¨older continuity for such weak solutions.

关键词： Caputo derivative Marchaud derivative porous medium equation Hlder continuity Nonlocal diffusion