To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem...To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Two oscillatory criteria of solutions for systems of parabolic differential equations with deviating arguments are obtained.展开更多
We study a class of semilinear periodic-parabolic boundary value problem with small diffusion coefficients,give its asymptotic behavior when the diffusion cofficients go to limit zero.
In this paper we study the decay estimate of global solutions to the initial-boundary value problem for double degenerate nonlinear parabolic equation by using a dif-ference inequality.
This paper deals with two parabolic initial-boundary value problems in multidimensional domain. The first problem describes the situation where the spherical medium is static and the nonlinear reaction takes place onl...This paper deals with two parabolic initial-boundary value problems in multidimensional domain. The first problem describes the situation where the spherical medium is static and the nonlinear reaction takes place only at a single point. We show that under some conditions, the solution blows up in finite time and the blow-up set is the whole spherical medium. When the spherical medium is allowed to move in a special space, we investigate another parabolic initial-boundary value problem. It is proved that the blow-up can be avoided if the acceleration of the motion satisfies certain conditions.展开更多
The homogenization of a class of degenerate quasilinear parabolic equations is studied. The Ap weight theory and the classical compensated compactness method are incorporated to obtain the homogenized equation.
In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t...In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t）=（u^p11v^p12）（1,t）,vx（1,t）=（u^p21v^p22）（1,t）,t∈（0,T）,u（x,0）=u0（x）,v（x,0）=v0（x）,x∈（0,1）We will prove that there exist two positive constants such that：c≤max x∈[0,1]u（x,t）（T-t）^r（l1-1）≤C,0〈t〈T,c≤max x∈[0,1] v（x,t）（T-t）^1/（t1-1）≤C,0〈t〈T.where l1=l21α/α2＋l22,r=α1/α2〉1,α1≤α2〈0.展开更多
In this paper we introduce an implementation for the efficient numericalsolution of exterior initial boundary value problem for parabolic equation. The problemis reformulated as an equivalent one on a boundary T using...In this paper we introduce an implementation for the efficient numericalsolution of exterior initial boundary value problem for parabolic equation. The problemis reformulated as an equivalent one on a boundary T using natural boundary reduction.The governing equation is first discretized in time, leading to a time-stepping scheme,where an exterior elliptic problem has to be solved in each time step. By Fourier ex-pansion, we derive a natural integral equation of the elliptic problem related to timestep and Poisson integral integral formula over exterior circular domain. Finite elementdiscretization of the natural integral equation is employed to solve this problem. Thecomputational aspects of this method are discussed. Numerical results are presented toillustrate feasibility and efficiency of our method.展开更多
Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new...Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.展开更多
文摘To study a class of boundary value problems of parabolic differential equations with deviating arguments, averaging technique, Green’s formula and symbol function sign(·) are used. The multi dimensional problem was reduced to a one dimensional oscillation problem for ordinary differential equations or inequalities. Two oscillatory criteria of solutions for systems of parabolic differential equations with deviating arguments are obtained.
文摘We study a class of semilinear periodic-parabolic boundary value problem with small diffusion coefficients,give its asymptotic behavior when the diffusion cofficients go to limit zero.
基金Supported by the NNSF of China(10441002)Supported by NNSF of Henan Province(200510466011)
文摘In this paper we study the decay estimate of global solutions to the initial-boundary value problem for double degenerate nonlinear parabolic equation by using a dif-ference inequality.
基金Supported by the Innovation Project for University Prominent Research Talents of Henan (2003KJCX008)
文摘This paper deals with two parabolic initial-boundary value problems in multidimensional domain. The first problem describes the situation where the spherical medium is static and the nonlinear reaction takes place only at a single point. We show that under some conditions, the solution blows up in finite time and the blow-up set is the whole spherical medium. When the spherical medium is allowed to move in a special space, we investigate another parabolic initial-boundary value problem. It is proved that the blow-up can be avoided if the acceleration of the motion satisfies certain conditions.
基金the Foundations of Returned Overseas Chinese Education Ministry and the Key Teachers Foundation of Chongqing University.
文摘The homogenization of a class of degenerate quasilinear parabolic equations is studied. The Ap weight theory and the classical compensated compactness method are incorporated to obtain the homogenized equation.
文摘In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t）=（u^p11v^p12）（1,t）,vx（1,t）=（u^p21v^p22）（1,t）,t∈（0,T）,u（x,0）=u0（x）,v（x,0）=v0（x）,x∈（0,1）We will prove that there exist two positive constants such that：c≤max x∈[0,1]u（x,t）（T-t）^r（l1-1）≤C,0〈t〈T,c≤max x∈[0,1] v（x,t）（T-t）^1/（t1-1）≤C,0〈t〈T.where l1=l21α/α2＋l22,r=α1/α2〉1,α1≤α2〈0.
基金National Natural Science Foundation of China(19701001)
文摘In this paper we introduce an implementation for the efficient numericalsolution of exterior initial boundary value problem for parabolic equation. The problemis reformulated as an equivalent one on a boundary T using natural boundary reduction.The governing equation is first discretized in time, leading to a time-stepping scheme,where an exterior elliptic problem has to be solved in each time step. By Fourier ex-pansion, we derive a natural integral equation of the elliptic problem related to timestep and Poisson integral integral formula over exterior circular domain. Finite elementdiscretization of the natural integral equation is employed to solve this problem. Thecomputational aspects of this method are discussed. Numerical results are presented toillustrate feasibility and efficiency of our method.
基金supported by the National Natural Science Foundation of China(No.11371297)the Science Foundation of Xiamen University of Technology(No.XYK201448)
文摘Consider the initial boundary value problem of the strong degenerate parabolic equation ?_(xx)u + u?_yu-?_tu = f(x, y, t, u),(x, y, t) ∈ Q_T = Ω×(0, T)with a homogeneous boundary condition. By introducing a new kind of entropy solution, according to Oleinik rules, the partial boundary condition is given to assure the well-posedness of the problem. By the parabolic regularization method, the uniform estimate of the gradient is obtained, and by using Kolmogoroff 's theorem, the solvability of the equation is obtained in BV(Q_T) sense. The stability of the solutions is obtained by Kruzkov's double variables method.
