In this article, the authors deal with the Cauchy problem of a nonlinear parabolic equation with variable density and absorption. By using energy methods, the authors prove that the interfaces can disappear in finite ...In this article, the authors deal with the Cauchy problem of a nonlinear parabolic equation with variable density and absorption. By using energy methods, the authors prove that the interfaces can disappear in finite time under some assumptions on the density functions.展开更多
This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = （σ ＋ m ）n / （ n-σ- 2 ） is its critical exponent provided ...This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = （σ ＋ m ）n / （ n-σ- 2 ） is its critical exponent provided max{-1, [ （1- m ）n- 2] / （ n ＋ l ） } 〈 σ 〈 n- 2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Futhermore, we demonstrate that if max（1, σ ＋ m） 〈 p 〈 pc, then every positive solution of the equations blows up in finite time; whereas for p 〉 pc, the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n〈σ＋2.展开更多
We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） an...We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） and without any restriction on the upper growth p of nonlinearity, except that p 〉 2, we show the existences of random attractor in D0^1,2（DN, σ） space, where DN is an arbitrary （bounded or unbounded） domain in R^N N 〉 2. For this purpose, some abstract results based on the omega-limit compactness are established.展开更多
In this article, it is shown that there exists a unique viscosity solution of the Cauchy problem for a degenerate parabolic equation with non-divergence form.
This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate pLaplace equation （p 〉 2）ut-div（b（x,t,u）｜↓△u｜^p-2↓△u）=f...This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate pLaplace equation （p 〉 2）ut-div（b（x,t,u）｜↓△u｜^p-2↓△u）=f（x,u,t）The initial-boundary value problem in a bounded domain of R^N is considered under mixed boundary conditions. The existence of local-in-time weak solutions is obtained.展开更多
The homogenization of one kind of nonlinear parabolic equation is studied. The weak convergence and corrector results are obtained by combining carefully the compactness method and two-scale convergence method in the ...The homogenization of one kind of nonlinear parabolic equation is studied. The weak convergence and corrector results are obtained by combining carefully the compactness method and two-scale convergence method in the homogenization theory.展开更多
This article discusses the existence and uniqueness of renormalized solutions for a class of degenerate parabolic equations b（u）t - div（a（u, u）） = H（u）（f ＋ divg）.
Abstract In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations...Abstract In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations, the blow-up rate determined by the interaction between the diffusion and the boundary flux is obtained. Compared with previous results, the gradient term, whose exponent does not exceed two, does not affect the blow-up rate of the solutions.展开更多
The existences of the global attractor Ae for a degenerate parabolic equation and of the homogenized attractor A0 for the corresponding homogenized equation are studied, and explicit estimates for the distance between...The existences of the global attractor Ae for a degenerate parabolic equation and of the homogenized attractor A0 for the corresponding homogenized equation are studied, and explicit estimates for the distance between Ae and A0 are given.展开更多
In this paper we discuss the quasilinear parabolic equation U_■=▽(u^u(1-u)~β.Vu)+■(x,t.u)▽u+C(x,t,u) which is degenerate at u =0 and u=1.Let u(x,t)be a weak solution of the equation satisfying 0<u(x,t)<1.Un...In this paper we discuss the quasilinear parabolic equation U_■=▽(u^u(1-u)~β.Vu)+■(x,t.u)▽u+C(x,t,u) which is degenerate at u =0 and u=1.Let u(x,t)be a weak solution of the equation satisfying 0<u(x,t)<1.Under some assumptions we establish H■lder continuity of u(x,t).展开更多
This paper studies the conditions of blow up in finite time of solutions of initial boundary value problem for a class nonlinear doubly degenerate parabolic equation.
In this paper, we prove the existence of solution of the Cauchy problem of a nonlinear degenerate parabolic equation. Moreover some regularizing effects are exhibited.
In this paper, we study the boundary value problem （BVP） for a degenerate parabolic equation. By introducing a proper notion of weak solutions, we prove the uniqueness and existence of weak solutions of the problem....In this paper, we study the boundary value problem （BVP） for a degenerate parabolic equation. By introducing a proper notion of weak solutions, we prove the uniqueness and existence of weak solutions of the problem. The localization of weak solutions is also discussed, which plays a key role in the proof of the uniqueness.展开更多
In this paper we study the strong and weak property of travelling wave front solutions for a class of degenerate parabolic equations. How the strong and weak property changes under the effects of wave speed and reacti...In this paper we study the strong and weak property of travelling wave front solutions for a class of degenerate parabolic equations. How the strong and weak property changes under the effects of wave speed and reaction diffusion terms are showed.展开更多
We study some classes of functions satisfying the assumptions similar to but weaker than those for the classical B2 function classes used in the research of quasi-linear parabolic equations as well as the ones used in...We study some classes of functions satisfying the assumptions similar to but weaker than those for the classical B2 function classes used in the research of quasi-linear parabolic equations as well as the ones used in the research of degenerate parabolic equations including porous medium equations. Consequently, we prove that a function in such a class is continuous. As an application, we obtain the estimate for the continuous modulus of the solutions of a few degenerate parabolic equations in divergence form, including the anisotropic porous equations.展开更多
In this paper, we consider the Cauchy problem of degenerate parabolic equation not in divergence form u, = uPAu ＋ uq, p 〉 1, q 〉 1, and give the blow-up conditions and the critical Fujita exponents for the existenc...In this paper, we consider the Cauchy problem of degenerate parabolic equation not in divergence form u, = uPAu ＋ uq, p 〉 1, q 〉 1, and give the blow-up conditions and the critical Fujita exponents for the existence of global and non-global solutions to the Cauchy problem.展开更多
The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in whic...The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.展开更多
In this paper,we study the existence and regularity of travelling wave front solutions for some degenerate parabolic equations (u^m/m)t=u_(xx)+u^nf(u),where m,n>0 and f(u)～1-u.We show that the existence and regula...In this paper,we study the existence and regularity of travelling wave front solutions for some degenerate parabolic equations (u^m/m)t=u_(xx)+u^nf(u),where m,n>0 and f(u)～1-u.We show that the existence and regularity of travelling wave front solutions depend on the parameters m,n and the wave speed c.展开更多
基金This work is supported in part by NNSF of China(10571126)in part by Program for New Century Excellent Talents in University
文摘In this article, the authors deal with the Cauchy problem of a nonlinear parabolic equation with variable density and absorption. By using energy methods, the authors prove that the interfaces can disappear in finite time under some assumptions on the density functions.
基金supported by the National Natural Science Foundations of China(10971061)Hunan Provincial Natural Science Foundation of China (09JJ6013)
文摘This article deals with the degenerate parabolic equations in exterior domains and with inhomogeneous Dirichlet boundary conditions. We obtain that pc = （σ ＋ m ）n / （ n-σ- 2 ） is its critical exponent provided max{-1, [ （1- m ）n- 2] / （ n ＋ l ） } 〈 σ 〈 n- 2. This critical exponent is not the same as that for the corresponding equations with the boundary value 0, but is more closely tied to the critical exponent of the elliptic type degenerate equations. Futhermore, we demonstrate that if max（1, σ ＋ m） 〈 p 〈 pc, then every positive solution of the equations blows up in finite time; whereas for p 〉 pc, the equations admit global positive solutions for some boundary values and initial data. Meantime, we also demonstrate that its positive solutions blow up in finite time provided n〈σ＋2.
基金supported by China NSF(11271388)Scientific and Technological Research Program of Chongqing Municipal Education Commission(KJ1400430)Basis and Frontier Research Project of Chongqing(cstc2014jcyj A00035)
文摘We study the regularity of random attractors for a class of degenerate parabolic equations with leading term div（o（x）↓△u） and multiplicative noises. Under some mild conditions on the diffusion variable o（x） and without any restriction on the upper growth p of nonlinearity, except that p 〉 2, we show the existences of random attractor in D0^1,2（DN, σ） space, where DN is an arbitrary （bounded or unbounded） domain in R^N N 〉 2. For this purpose, some abstract results based on the omega-limit compactness are established.
基金Supported in part by Dalian Nationalities University (20076209)Departmentof Education of Liaoning Province (2009A152)National Natural Science Foundation of China (10471156,10901030)
文摘In this article, it is shown that there exists a unique viscosity solution of the Cauchy problem for a degenerate parabolic equation with non-divergence form.
基金The NSFC(10371050)and the"985"program of Jilin University.
文摘This paper deals with a class of doubly degenerate parabolic equations, including as particular cases the porous medium equation and the degenerate pLaplace equation （p 〉 2）ut-div（b（x,t,u）｜↓△u｜^p-2↓△u）=f（x,u,t）The initial-boundary value problem in a bounded domain of R^N is considered under mixed boundary conditions. The existence of local-in-time weak solutions is obtained.
基金Funded by the Scientific Research Foundation for Returned Overseas Chinese Scholars under the State Education Ministry, and the Key Teachers’ Foundation of Chongqing University.
文摘The homogenization of one kind of nonlinear parabolic equation is studied. The weak convergence and corrector results are obtained by combining carefully the compactness method and two-scale convergence method in the homogenization theory.
基金supported by Science Foundation of Xiamen University of Technology (YKJ08020R)
文摘This article discusses the existence and uniqueness of renormalized solutions for a class of degenerate parabolic equations b（u）t - div（a（u, u）） = H（u）（f ＋ divg）.
基金Project supported by the Youth Foundation of the National Natural Science Foundation of China(No. 10701061)
文摘Abstract In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations, the blow-up rate determined by the interaction between the diffusion and the boundary flux is obtained. Compared with previous results, the gradient term, whose exponent does not exceed two, does not affect the blow-up rate of the solutions.
文摘The global existence and finite time blow up of the positive solution for a nonlinear degenerate parabolic equation with non- local source are studied
基金the Scientific Research Foundation for Returned Overseas Chinese Scholars under the State Education Ministrythe Key Teachers’Foundation of Chongqing University
文摘The existences of the global attractor Ae for a degenerate parabolic equation and of the homogenized attractor A0 for the corresponding homogenized equation are studied, and explicit estimates for the distance between Ae and A0 are given.
文摘In this paper we discuss the quasilinear parabolic equation U_■=▽(u^u(1-u)~β.Vu)+■(x,t.u)▽u+C(x,t,u) which is degenerate at u =0 and u=1.Let u(x,t)be a weak solution of the equation satisfying 0<u(x,t)<1.Under some assumptions we establish H■lder continuity of u(x,t).
文摘This paper studies the conditions of blow up in finite time of solutions of initial boundary value problem for a class nonlinear doubly degenerate parabolic equation.
文摘In this paper, we prove the existence of solution of the Cauchy problem of a nonlinear degenerate parabolic equation. Moreover some regularizing effects are exhibited.
基金Young Teachers Foundation (420010302318) of Jilin University
文摘In this paper, we study the boundary value problem （BVP） for a degenerate parabolic equation. By introducing a proper notion of weak solutions, we prove the uniqueness and existence of weak solutions of the problem. The localization of weak solutions is also discussed, which plays a key role in the proof of the uniqueness.
文摘In this paper we study the strong and weak property of travelling wave front solutions for a class of degenerate parabolic equations. How the strong and weak property changes under the effects of wave speed and reaction diffusion terms are showed.
基金This work was partially supported by the University of Nancy Iby National 973-Project from MOST as well as well Trans-Century Training Programme Foundation for the Talents by Ministry of Education
文摘We study some classes of functions satisfying the assumptions similar to but weaker than those for the classical B2 function classes used in the research of quasi-linear parabolic equations as well as the ones used in the research of degenerate parabolic equations including porous medium equations. Consequently, we prove that a function in such a class is continuous. As an application, we obtain the estimate for the continuous modulus of the solutions of a few degenerate parabolic equations in divergence form, including the anisotropic porous equations.
文摘In this paper, we consider the Cauchy problem of degenerate parabolic equation not in divergence form u, = uPAu ＋ uq, p 〉 1, q 〉 1, and give the blow-up conditions and the critical Fujita exponents for the existence of global and non-global solutions to the Cauchy problem.
基金supported by the National Natural Science Foundation of China(Nos.11061018,11261029)the Youth Foundation of Lanzhou Jiaotong University(No.2011028)+1 种基金the Long Yuan Young Creative Talents Support Program(No.252003)the Joint Funds of the Gansu Provincial Natural Science Foundation of China(No.1212RJZA043)
文摘The authors investigate an inverse problem of determining the radiative coefficient in a degenerate parabolic equation from the final overspecified data. Being different from other inverse coefficient problems in which the principle coefficients are assumed to be strictly positive definite, the mathematical model discussed in this paper belongs to the second order parabolic equations with non-negative characteristic form, namely, there exists a degeneracy on the lateral boundaries of the domain. Based on the optimal control framework, the problem is transformed into an optimization problem and the existence of the minimizer is established. After the necessary conditions which must be satisfied by the minimizer are deduced, the uniqueness and stability of the minimizer are proved. By minor modification of the cost functional and some a priori regularity conditions imposed on the forward operator, the convergence of the minimizer for the noisy input data is obtained in this paper. The results can be extended to more general degenerate parabolic equations.
基金This project is supported by the Notional Natural Science Foundation of China
文摘In this paper,we study the existence and regularity of travelling wave front solutions for some degenerate parabolic equations (u^m/m)t=u_(xx)+u^nf(u),where m,n>0 and f(u)～1-u.We show that the existence and regularity of travelling wave front solutions depend on the parameters m,n and the wave speed c.
