Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we pro...Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.展开更多
In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f（u）xx,x∈Ω,t〉0,u（x,0）=u0（x）,ut（x,0）=u1（x）,x∈Ω,u（0,t）=u（1,t）=0,t≥0, where Ω...In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f（u）xx,x∈Ω,t〉0,u（x,0）=u0（x）,ut（x,0）=u1（x）,x∈Ω,u（0,t）=u（1,t）=0,t≥0, where Ω = （0, 1）. First, we obtain the existence of local Wkp solutions. Then, we prove that, if f（s） ∈ Ck＋1（R） is nondecreasing, f（0） = 0 and ｜f（u）｜ ≤ C1｜u｜∫0uf（s）ds ＋ C2, u0（x）,u1（x） ∈ WkP（Ω） ∩ W01,P（Ω）, k≥ 1, 1 〈 p≤ ∞, then for any T 〉 0 the problem admits a unique solution u（x, t） ∈ W2,∞（0, T; WkP（Ω） ∩ W01,P（Ω））. Finally, the finite time blow-up of solutions and global Wkp solution of generalized IMBo equations are discussed.展开更多
In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local exist...In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.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
This paper deals with an initial-boundary value problem of a fourth-order parabolic equation involving Logarithmic type p-Laplacian,which could be proposed as a model for the epitaxial growth of thin films.By using th...This paper deals with an initial-boundary value problem of a fourth-order parabolic equation involving Logarithmic type p-Laplacian,which could be proposed as a model for the epitaxial growth of thin films.By using the variational method and the logarithmic type Sobolev inequality,we give some threshold results for blow-up solutions and global solutions,which could be classified by the initial energy.The asymptotic estimates about blow-up time and decay estimate of weak solutions are obtained.展开更多
In this paper,by using the coincidence degree theory,the existence of solutions for a coupled system of fractional p-Laplacian differential equations at resonance is studied.The result obtained in this paper extends s...In this paper,by using the coincidence degree theory,the existence of solutions for a coupled system of fractional p-Laplacian differential equations at resonance is studied.The result obtained in this paper extends some known results.An example is given to illustrate our result.展开更多
This paper deals with the blow-up properties of positive solutions to a localized degenerate and singular parabolic equation with weighted nonlocal boundary condi- tions. Under appropriate hypotheses, the global exist...This paper deals with the blow-up properties of positive solutions to a localized degenerate and singular parabolic equation with weighted nonlocal boundary condi- tions. Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. Furthermore, the global blow-up behavior and the uniform blow-up profile of blow-up solutions are also described. We find that the blow-up set is the whole domain {0, a}, including the boundaries, and this differs from parabolic equations with local sources case or with homogeneous Dirichlet boundary conditions case.展开更多
基金supported by the Beijing Natural Science Foundation(1212003)。
文摘Let?denote a smooth,bounded domain in R^(N)(N≥2).Suppose that g is a nondecreasing C^(1)positive function and assume that b(x)is continuous and nonnegative inΩ,and that it may be singular on■Ω.In this paper,we provide sufficient and necessary conditions on the existence of boundary blow-up solutions to the p-Laplacian problem△_(p)u=b(x)g(u)for x∈Ω,u(x)→+∞as dist(x,■Ω)→0.The estimates of such solutions are also investigated.Moreover,when b has strong singularity,the nonexistence of boundary blow-up(radial)solutions and infinitely many radial solutions are also considered.
基金supported by National Natural Science Foundation of China(10871055,10926149)Natural Science Foundation of Heilongjiang Province (A2007-02+2 种基金A200810)Science and Technology Foundation of Education Office of Heilongjiang Province(11541276)Foundational Science Founda-tion of Harbin Engineering University
文摘In this article, we study the initial boundary value problem of generalized Pochhammer-Chree equation utt-uxx-uxxt-uxxtt=f（u）xx,x∈Ω,t〉0,u（x,0）=u0（x）,ut（x,0）=u1（x）,x∈Ω,u（0,t）=u（1,t）=0,t≥0, where Ω = （0, 1）. First, we obtain the existence of local Wkp solutions. Then, we prove that, if f（s） ∈ Ck＋1（R） is nondecreasing, f（0） = 0 and ｜f（u）｜ ≤ C1｜u｜∫0uf（s）ds ＋ C2, u0（x）,u1（x） ∈ WkP（Ω） ∩ W01,P（Ω）, k≥ 1, 1 〈 p≤ ∞, then for any T 〉 0 the problem admits a unique solution u（x, t） ∈ W2,∞（0, T; WkP（Ω） ∩ W01,P（Ω））. Finally, the finite time blow-up of solutions and global Wkp solution of generalized IMBo equations are discussed.
基金supported by a grant from the National High Technology Researchand and Development Program of China (863 Program) (2009AA044501)by NSFC (10776035+2 种基金10771085)by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Educationby the 985 program of Jilin University
文摘In this paper, the authors discuss the global existence and blow-up of the solution to an evolution p-Laplace system with nonlinear sources and nonlinear boundary condition. The authors first establish the local existence of solutions, then give a necessary and sufficient condition on the global existence of the positive solution.
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 Shandong Provincial Natural Science Foundation of China(Grant No.ZR2021MA003).
文摘This paper deals with an initial-boundary value problem of a fourth-order parabolic equation involving Logarithmic type p-Laplacian,which could be proposed as a model for the epitaxial growth of thin films.By using the variational method and the logarithmic type Sobolev inequality,we give some threshold results for blow-up solutions and global solutions,which could be classified by the initial energy.The asymptotic estimates about blow-up time and decay estimate of weak solutions are obtained.
基金The Key NSF(KJ2015A196)of Anhui Higher Educationthe Young Foundation(2015QN19)of Hefei Normal University
文摘In this paper,by using the coincidence degree theory,the existence of solutions for a coupled system of fractional p-Laplacian differential equations at resonance is studied.The result obtained in this paper extends some known results.An example is given to illustrate our result.
基金supported by the research scheme of the natural science of the universities of Jiangsu province(08KJD110017 and 13KJB110028)
文摘This paper deals with the blow-up properties of positive solutions to a localized degenerate and singular parabolic equation with weighted nonlocal boundary condi- tions. Under appropriate hypotheses, the global existence and finite time blow-up of positive solutions are obtained. Furthermore, the global blow-up behavior and the uniform blow-up profile of blow-up solutions are also described. We find that the blow-up set is the whole domain {0, a}, including the boundaries, and this differs from parabolic equations with local sources case or with homogeneous Dirichlet boundary conditions case.
