This article concerns the existence of weak solutions of the first boundary value problem for a kind of strongly degenerate quasilinear parabolic equation in the anisotropic Sobolev Space. With the theory of anisotrop...This article concerns the existence of weak solutions of the first boundary value problem for a kind of strongly degenerate quasilinear parabolic equation in the anisotropic Sobolev Space. With the theory of anisotropic Sobolev spaces an existence result is proved.展开更多
We deal with the existence of periodic solutions for doubly degenerate quasilinear parabolic equations of higher order,which can degenerate,on a part of the boundary,on a segment in the interior of the domain and in t...We deal with the existence of periodic solutions for doubly degenerate quasilinear parabolic equations of higher order,which can degenerate,on a part of the boundary,on a segment in the interior of the domain and in time.展开更多
The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form： μt-△φ（μ） = 0 ,where φ ε C1（R^1） is a strictly monotone increasing function. Cle...The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form： μt-△φ（μ） = 0 ,where φ ε C1（R^1） is a strictly monotone increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of φ＇（.） is permitted to have zero measure. This is an answer to an open problem in [13, p. 288].展开更多
A combination of the classical Newton Method and the multigrid method, i.e., a Newton multigrid method is given for solving quasilinear parabolic equations discretized by finite elements. The convergence of the algori...A combination of the classical Newton Method and the multigrid method, i.e., a Newton multigrid method is given for solving quasilinear parabolic equations discretized by finite elements. The convergence of the algorithm is obtained for only one step Newton iteration per level. The asymptotically computational cost for quasilinear parabolic problems is O（NNk） similar to multigrid method for linear parabolic problems.展开更多
Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.
In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|^(p-2)?u) =-|u|^(β-1) u + α|u|^(q-2 )u,where p > 1, β > 0, q≥1 and α > 0. By using Gagliard...In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|^(p-2)?u) =-|u|^(β-1) u + α|u|^(q-2 )u,where p > 1, β > 0, q≥1 and α > 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.展开更多
This paper deals with the existence and nonexistence of global positive solutions of the following quasilinear parabolic equations:u t=1m△u m-u n, x∈Ω,t>0 1m·u mv=u p,x∈Ω,t>0 u(x,0)=u 0(x)>0,x∈Ω-w...This paper deals with the existence and nonexistence of global positive solutions of the following quasilinear parabolic equations:u t=1m△u m-u n, x∈Ω,t>0 1m·u mv=u p,x∈Ω,t>0 u(x,0)=u 0(x)>0,x∈Ω-where Ω∈R N is a bounded domain with smooth boundary Ω,m,n,p are positive constants, γ is the outward normal vector. The necessary and sufficient conditions for the global existence of solutions are obtained.展开更多
We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform c...We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L2 norm is proved and numerical examples are presented.展开更多
The aim of this paper is to discuss the Cauchy problem for quasilinear degenerate parabolic equations of the formwhere φ∈C1(R1) is a strictly monotonically increasing function. Clearly, the above equation has strong...The aim of this paper is to discuss the Cauchy problem for quasilinear degenerate parabolic equations of the formwhere φ∈C1(R1) is a strictly monotonically increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of φ'(-) is permitted to have zero measure. In particular, the existence of interfaces of solutions is obtained.展开更多
This paper investigates the qualitative properties of solutions to certain quasilinear parabolic equations. Under appropriate conditions, we obtain that the solution either exists globally or blows up in finite time b...This paper investigates the qualitative properties of solutions to certain quasilinear parabolic equations. Under appropriate conditions, we obtain that the solution either exists globally or blows up in finite time by making use of the energy method and subsolution techniques. We find out that the behavior of solution heavily depends on the sign and the growth rate of the nonlinear reaction term and the nonlinear flux through boundary at infinity.展开更多
The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping u_(tt)+u_(t)-div(a(u)u)=0,and show that,at least when n≥3,they tend,as t-+∞,to those of the nonlin...The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping u_(tt)+u_(t)-div(a(u)u)=0,and show that,at least when n≥3,they tend,as t-+∞,to those of the nonlinear parabolic equation v_t-div(a(v)v)=0,in the sense that the norm||u(.,t)-v(.,t)||_(L∞(R^n))of the difference u-v decays faster than that of either u or v.This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves,first observed by Hsiao,L.and Liu Taiping(see[1,2]).展开更多
This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of so...This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vázquez and the comparison principle, we deduce that the blow-up occurs only on the boundary ?Ω. In addition, for a bounded Lipschitz domain Ω, we establish the blow-up rate estimates for the positive solution to this problem with a = 0.展开更多
In this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of local solutions of quasilinear parabolic equation with critical Sobolev exponent and with lower energy ini...In this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of local solutions of quasilinear parabolic equation with critical Sobolev exponent and with lower energy initial value; we also describe the asymptotic behavior of global solutions with high energy initial value.展开更多
Blow-up behaviour for the fourth-order quasilinear porous medium equation with source,ut=-（｜u｜^nu）xxxx＋｜u｜^p-1u in R×R＋,where n 〉 0, p 〉 1, is studied. Countable and finite families of similarity blow-u...Blow-up behaviour for the fourth-order quasilinear porous medium equation with source,ut=-（｜u｜^nu）xxxx＋｜u｜^p-1u in R×R＋,where n 〉 0, p 〉 1, is studied. Countable and finite families of similarity blow-up patterns of the form us（x,t）=（T-t）^-1/p-1f（y）,where y=x/（T-t）^β＇ β=p-（n＋1）/4（p-1）,which blow-up as t → T^- 〈∞ are described. These solutions explain key features of regional （for p = n＋1）, single point （for p 〉 n＋1）, and global （for p ∈ （1,n＋1））blowup. The concepts and various variational, bifurcation, and numerical approaches for revealing the structure and multiplicities of such blow-up patterns are presented.展开更多
We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-,two-and three dimensional unbounded domains.The exact artificial boundary conditions are obtained on the arti...We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-,two-and three dimensional unbounded domains.The exact artificial boundary conditions are obtained on the artificial boundaries.Then the original problems are reduced to equivalent problems in bounded domains.A fi-nite difference method is applied to solve the reduced problems,and some numerical examples are provided to show the effectiveness of the method.展开更多
基金The project is supported by NNSF of China (10371116)
文摘This article concerns the existence of weak solutions of the first boundary value problem for a kind of strongly degenerate quasilinear parabolic equation in the anisotropic Sobolev Space. With the theory of anisotropic Sobolev spaces an existence result is proved.
基金Project supported partially by NNSF of China Grant No.10171008NSF of Hunan Province Grant No.03JJY3003
文摘We deal with the existence of periodic solutions for doubly degenerate quasilinear parabolic equations of higher order,which can degenerate,on a part of the boundary,on a segment in the interior of the domain and in time.
基金Project supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Education Institutions of MOE(No.[2000]26)the 973 Project of the Ministry of Science and Technology of China(No.2006CB805902)+1 种基金the National Natural Science Foundation of China(No.10571072)the Key Laboratory of Symbolic Computation and Knowledge Engineering of the Ministry of Education of China and the 985 Project of Jilin University.
文摘The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form： μt-△φ（μ） = 0 ,where φ ε C1（R^1） is a strictly monotone increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of φ＇（.） is permitted to have zero measure. This is an answer to an open problem in [13, p. 288].
基金This research is supported by the National Natural Science Foundation of China(10471011).
文摘A combination of the classical Newton Method and the multigrid method, i.e., a Newton multigrid method is given for solving quasilinear parabolic equations discretized by finite elements. The convergence of the algorithm is obtained for only one step Newton iteration per level. The asymptotically computational cost for quasilinear parabolic problems is O（NNk） similar to multigrid method for linear parabolic problems.
文摘Under some conditions, one seows that the generalized solutions of the first boundary value problem for the equation [GRAPHICS] have the property of finite speed of propagation.
基金supported by National Natural Science Foundation of China (Grant Nos. 11371286 and 11401458)the Special Fund of Education Department (Grant No. 2013JK0586)the Youth Natural Science Grant of Shaanxi Province of China (Grant No. 2013JQ1015)
文摘In this paper, we mainly consider the initial boundary problem for a quasilinear parabolic equation u_t-div(|?u|^(p-2)?u) =-|u|^(β-1) u + α|u|^(q-2 )u,where p > 1, β > 0, q≥1 and α > 0. By using Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blowup and extinction are classified completely in the different ranges of reaction exponents.
文摘This paper deals with the existence and nonexistence of global positive solutions of the following quasilinear parabolic equations:u t=1m△u m-u n, x∈Ω,t>0 1m·u mv=u p,x∈Ω,t>0 u(x,0)=u 0(x)>0,x∈Ω-where Ω∈R N is a bounded domain with smooth boundary Ω,m,n,p are positive constants, γ is the outward normal vector. The necessary and sufficient conditions for the global existence of solutions are obtained.
文摘We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L2 norm is proved and numerical examples are presented.
基金Project supported by the Teaching and Research Award Found for Outstanding Young Teachers in Higher Education Institutions of MOE ([2000]26)China and the NNSF (1001015) of China
文摘The aim of this paper is to discuss the Cauchy problem for quasilinear degenerate parabolic equations of the formwhere φ∈C1(R1) is a strictly monotonically increasing function. Clearly, the above equation has strong degeneracy, i.e., the set of zero points of φ'(-) is permitted to have zero measure. In particular, the existence of interfaces of solutions is obtained.
基金This work was supported by China Postdoctoral Science Foundation.
文摘This paper investigates the qualitative properties of solutions to certain quasilinear parabolic equations. Under appropriate conditions, we obtain that the solution either exists globally or blows up in finite time by making use of the energy method and subsolution techniques. We find out that the behavior of solution heavily depends on the sign and the growth rate of the nonlinear reaction term and the nonlinear flux through boundary at infinity.
文摘The authors consider the asymptotic behavior of solutions of the quasilinear hyperbolic equation with linear damping u_(tt)+u_(t)-div(a(u)u)=0,and show that,at least when n≥3,they tend,as t-+∞,to those of the nonlinear parabolic equation v_t-div(a(v)v)=0,in the sense that the norm||u(.,t)-v(.,t)||_(L∞(R^n))of the difference u-v decays faster than that of either u or v.This provides another example of the diffusion phenomenon of nonlinear hyperbolic waves,first observed by Hsiao,L.and Liu Taiping(see[1,2]).
基金This work was supported by the National Natural Science Foundation of China (Grant Nos. 10471022 and 10601011)Key Project of the Ministry of Education of China (Grant No. 104090)
文摘This paper deals with the properties of positive solutions to a quasilinear parabolic equation with the nonlinear absorption and the boundary flux. The necessary and sufficient conditions on the global existence of solutions are described in terms of different parameters appearing in this problem. Moreover, by a result of Chasseign and Vázquez and the comparison principle, we deduce that the blow-up occurs only on the boundary ?Ω. In addition, for a bounded Lipschitz domain Ω, we establish the blow-up rate estimates for the positive solution to this problem with a = 0.
基金Supported by NSF(No:10171083 and 10371021)of China Laboratory of Mathematics for Nonlinear Sciences of Fudan University
文摘In this paper we consider the existence and asymptotic estimates of global solutions and finite time blowup of local solutions of quasilinear parabolic equation with critical Sobolev exponent and with lower energy initial value; we also describe the asymptotic behavior of global solutions with high energy initial value.
文摘Blow-up behaviour for the fourth-order quasilinear porous medium equation with source,ut=-（｜u｜^nu）xxxx＋｜u｜^p-1u in R×R＋,where n 〉 0, p 〉 1, is studied. Countable and finite families of similarity blow-up patterns of the form us（x,t）=（T-t）^-1/p-1f（y）,where y=x/（T-t）^β＇ β=p-（n＋1）/4（p-1）,which blow-up as t → T^- 〈∞ are described. These solutions explain key features of regional （for p = n＋1）, single point （for p 〉 n＋1）, and global （for p ∈ （1,n＋1））blowup. The concepts and various variational, bifurcation, and numerical approaches for revealing the structure and multiplicities of such blow-up patterns are presented.
基金National Natural Science Foundation of China,Hong Kong Research Grants Council and FRG of Hong Kong Baptist University.
文摘We propose an artificial boundary method for solving the deterministic Kardar-Parisi-Zhang equation in one-,two-and three dimensional unbounded domains.The exact artificial boundary conditions are obtained on the artificial boundaries.Then the original problems are reduced to equivalent problems in bounded domains.A fi-nite difference method is applied to solve the reduced problems,and some numerical examples are provided to show the effectiveness of the method.
