consider a quasilinear parabolic equation in a band domain with inhomogeneous and unbounded boundary conditions.We show that,under certain conditions,the solution u of the initialboundary value problem tends to infini...consider a quasilinear parabolic equation in a band domain with inhomogeneous and unbounded boundary conditions.We show that,under certain conditions,the solution u of the initialboundary value problem tends to infinite as t→∞.Moreover,by using the zero number argument we show that for any x≠O,u_(x)(x,t)also tends as t→∞to infinity,that is,the gradient is asymptotically unbounded.展开更多
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
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
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.
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
In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one wea...In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one weak solution in a suitable weighted Sobolev space.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
In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.
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
This paper is devoted to study the classification of self-similar solutions to the quasilinear parabolic equations with nonlinear gradient termsu t = Δ(u m) - uq∣Δu∣ p withm ≥ 1,p,q > 0 andp +q >m. Form = 1...This paper is devoted to study the classification of self-similar solutions to the quasilinear parabolic equations with nonlinear gradient termsu t = Δ(u m) - uq∣Δu∣ p withm ≥ 1,p,q > 0 andp +q >m. Form = 1, it is shown that the very singular self-similar solution exists if and only ifnq + (n +1)p <n + 2, and in case of existence, such solution is unique. Form > 1, it is shown that very singular self-similar solutions exist if and only if 1 <m < 2 andnq + (n + 1)p < 2 +mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.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
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
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
基金The National Natural Science Foundation of China(12001375)。
文摘consider a quasilinear parabolic equation in a band domain with inhomogeneous and unbounded boundary conditions.We show that,under certain conditions,the solution u of the initialboundary value problem tends to infinite as t→∞.Moreover,by using the zero number argument we show that for any x≠O,u_(x)(x,t)also tends as t→∞to infinity,that is,the gradient is asymptotically unbounded.
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.
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.
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 research is supported by the Natural science Foundation of Hunan province
文摘In this paper,we consider nonlinear degenerate quasilinear parabolic initial boundary value problems of second order.Using results from the theory of pseudomonotone operators,we show that there exists at least one weak solution in a suitable weighted Sobolev space.
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Supported by the funds of the State Educational Commission of China for returned scholars from abroad
文摘In the paper, existence results for degenerate parabolic boundary value problems of higher order are proved. The weak solution is sought in a suitable weighted Sobolev space by using the generalized degree theory.
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 work was supported by the National Natural Science Foundation of China(Grant No.19831060)the"333"Project of Jiangsu Province.
文摘This paper is devoted to study the classification of self-similar solutions to the quasilinear parabolic equations with nonlinear gradient termsu t = Δ(u m) - uq∣Δu∣ p withm ≥ 1,p,q > 0 andp +q >m. Form = 1, it is shown that the very singular self-similar solution exists if and only ifnq + (n +1)p <n + 2, and in case of existence, such solution is unique. Form > 1, it is shown that very singular self-similar solutions exist if and only if 1 <m < 2 andnq + (n + 1)p < 2 +mn, and such solutions have compact support if they exist. Moreover, the interface relation is obtained.
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.
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.
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.
