An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary conditio...An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary condition is given for the solutions of the parameter estimatioll problem.展开更多
Let G =(V, E) be a locally finite connected weighted graph, and ? be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ?u + f(u) on G. The blow-up p...Let G =(V, E) be a locally finite connected weighted graph, and ? be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ?u + f(u) on G. The blow-up phenomenons for ut = ?u + f(u) are discussed in terms of two cases:(i) an initial condition is given;(ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.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
Error estimates are established for the finite element methods to solve a class of second order nonlinear parabolic equations. Optimal rates of convergence in L 2 and H 1 norms are derived.Meanwhile,the schemes are se...Error estimates are established for the finite element methods to solve a class of second order nonlinear parabolic equations. Optimal rates of convergence in L 2 and H 1 norms are derived.Meanwhile,the schemes are second order correct in time.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).
In this paper, we study the threshold result for the initial boundary value problem of non-homogeneous semilinear parabolic equations {μt-△μ=g(μ)+λf(x),(x,t)∈Ω×(0,T),μ=0,(x,t)∈ Ω×[0,T)...In this paper, we study the threshold result for the initial boundary value problem of non-homogeneous semilinear parabolic equations {μt-△μ=g(μ)+λf(x),(x,t)∈Ω×(0,T),μ=0,(x,t)∈ Ω×[0,T),μ(x,0)=μ0(x)≥0,x∈Ω.By combining a priori estimate of global solution with property of stationary solution set of problem (P), we prove that the minimal stationary solution Uλ(x) of problem (P) is stable, whereas, any other stationary solution is an initial datum threshold for the existence and nonexistence of global solution to problem (P).展开更多
In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner t...In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality,the Poincare′inequality,and the Hardy inequality.Then,by using the potential well method and the inequality mentioned above,we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy.Significantly,the relation between the above two phenomena is derived as a sharp condition.Moreover,we show that the global existence also holds for the case of a potential well family.展开更多
We study the Dirichlet problem associated to strongly nonlinear parabolic equations involving p(x) structure in W;L;(Q). We prove the existence of weak solutions by applying Galerkin’s approximation method.
We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, ...We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.展开更多
In this article,we investigate the longtime behavior for the following nonautonomous nonclassical parabolic equations on unbounded domain ut−∆ut−∆u+λu=f(x,u(x,t−ρ(t)))+g(x,t).Under some suitable conditions on the d...In this article,we investigate the longtime behavior for the following nonautonomous nonclassical parabolic equations on unbounded domain ut−∆ut−∆u+λu=f(x,u(x,t−ρ(t)))+g(x,t).Under some suitable conditions on the delay term f and the non-autonomous forcing term g,we prove the existence of uniform attractors in Banach space CH1(RN)for the multivalued process generated by non-autonomous nonclassical parabolic equations with delays in unbounded domain.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 authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div((|u|^δ + d0)|↓△|^p(x,t)-2↓△u) + f(x, t) (0 〈 δ 〈 2). They a...The authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div((|u|^δ + d0)|↓△|^p(x,t)-2↓△u) + f(x, t) (0 〈 δ 〈 2). They apply the method of parabolic regularization and Galerkin's method to prove the existence of solutions to the mentioned problem and then prove the uniqueness of the weak solution by arguing by contradiction. The authors prove that the solution approaches 0 in L^2 (Ω) norm as t →∞.展开更多
In this paper, we extend the applications of proper orthogonal decomposition (POD) method, i.e., apply POD method to a mixed finite element (MFE) formulation naturally satisfied Brezz-Babu^ka for parabolic equatio...In this paper, we extend the applications of proper orthogonal decomposition (POD) method, i.e., apply POD method to a mixed finite element (MFE) formulation naturally satisfied Brezz-Babu^ka for parabolic equations, establish a reduced-order MFE formulation with lower dimensions and sufficiently high accuracy, and provide the error estimates between the reduced-order POD MFE solutions and the classical MFE solutions and the implementation of algorithm for solving reduced-order MFE formulation. Some numerical examples illustrate the fact that the results of numerical computation are consis- tent with theoretical conclusions. Moreover, it is shown that the new reduced-order MFE formulation based on POD method is feasible and efficient for solving MFE formulation for parabolic equations.展开更多
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
In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations (△-e/et)u(x,t)+q(x,t)u^p(x,t)=0 and nonlinear parabolic equations (△-e/et)u(x,...In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations (△-e/et)u(x,t)+q(x,t)u^p(x,t)=0 and nonlinear parabolic equations (△-e/et)u(x,t)+h(x,t)u^p(x,t)=0(p 〉 1) on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang ([1], Bull. London Math. Soc. 38(2006), 1045-1053) and the author ([2], Nonlinear Anal. 74 (2011), 5141-5146).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
The author studies semilinear parabolic equations with initial and periodic boundary value conditions. In the presence of non-well-ordered sub- and super-solutions: "subsolution ≮ supersolution', the existence and...The author studies semilinear parabolic equations with initial and periodic boundary value conditions. In the presence of non-well-ordered sub- and super-solutions: "subsolution ≮ supersolution', the existence and stability/instability of equilibrium solutions are obtained.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 presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite differenc...This paper presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.展开更多
基金the post-doctoral funds of China and funds of State Educational Commission of China for returned scholars from abroad
文摘An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L1 setting. A necessary condition is given for the solutions of the parameter estimatioll problem.
基金supported by the National Science Foundation of China(11671401)supported by the Fundamental Research Funds for the Central Universitiesthe Research Funds of Renmin University of China(17XNH106)
文摘Let G =(V, E) be a locally finite connected weighted graph, and ? be the usual graph Laplacian. In this article, we study blow-up problems for the nonlinear parabolic equation ut = ?u + f(u) on G. The blow-up phenomenons for ut = ?u + f(u) are discussed in terms of two cases:(i) an initial condition is given;(ii) a Dirichlet boundary condition is given. We prove that if f satisfies appropriate conditions, then the corresponding solutions will blow up in a finite time.
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National Natural Science Foundation and Doctoral Foundation of Education Ministry of Stat
文摘Error estimates are established for the finite element methods to solve a class of second order nonlinear parabolic equations. Optimal rates of convergence in L 2 and H 1 norms are derived.Meanwhile,the schemes are second order correct in time.
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supported by Natural Science Foundation of China(10971061)Hunan Provincial Innovation Foundation For Postgraduate(CX2010B209)
文摘In this paper, we study the threshold result for the initial boundary value problem of non-homogeneous semilinear parabolic equations {μt-△μ=g(μ)+λf(x),(x,t)∈Ω×(0,T),μ=0,(x,t)∈ Ω×[0,T),μ(x,0)=μ0(x)≥0,x∈Ω.By combining a priori estimate of global solution with property of stationary solution set of problem (P), we prove that the minimal stationary solution Uλ(x) of problem (P) is stable, whereas, any other stationary solution is an initial datum threshold for the existence and nonexistence of global solution to problem (P).
文摘In this article,we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities.Firstly,we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality,the Poincare′inequality,and the Hardy inequality.Then,by using the potential well method and the inequality mentioned above,we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy.Significantly,the relation between the above two phenomena is derived as a sharp condition.Moreover,we show that the global existence also holds for the case of a potential well family.
文摘We study the Dirichlet problem associated to strongly nonlinear parabolic equations involving p(x) structure in W;L;(Q). We prove the existence of weak solutions by applying Galerkin’s approximation method.
基金Project supported by the National Natural Science Foundation of China(Grant No.11471262)the National Basic Research Program of China(Grant No.2012CB025904)the State Key Laboratory of Science and Engineering Computing and the Center for High Performance Computing of Northwestern Polytechnical University,China
文摘We study the hyperbolic–parabolic equations with rapidly oscillating coefficients. The formal second-order two-scale asymptotic expansion solutions are constructed by the multiscale asymptotic analysis. In addition, we theoretically explain the importance of the second-order two-scale solution by the error analysis in the pointwise sense. The associated explicit convergence rates are also obtained. Then a second-order two-scale numerical method based on the Newmark scheme is presented to solve the equations. Finally, some numerical examples are used to verify the effectiveness and efficiency of the multiscale numerical algorithm we proposed.
基金Supported by the 2018 research funding of higher education of Gansu province project[2018B-075]
文摘In this article,we investigate the longtime behavior for the following nonautonomous nonclassical parabolic equations on unbounded domain ut−∆ut−∆u+λu=f(x,u(x,t−ρ(t)))+g(x,t).Under some suitable conditions on the delay term f and the non-autonomous forcing term g,we prove the existence of uniform attractors in Banach space CH1(RN)for the multivalued process generated by non-autonomous nonclassical parabolic equations with delays in unbounded domain.
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 NSFC (10771085)Graduate Innovation Fund of Jilin University(20111034)the 985 program of Jilin University
文摘The authors of this article study the existence and uniqueness of weak so- lutions of the initial-boundary value problem for ut = div((|u|^δ + d0)|↓△|^p(x,t)-2↓△u) + f(x, t) (0 〈 δ 〈 2). They apply the method of parabolic regularization and Galerkin's method to prove the existence of solutions to the mentioned problem and then prove the uniqueness of the weak solution by arguing by contradiction. The authors prove that the solution approaches 0 in L^2 (Ω) norm as t →∞.
基金supported by the National Science Foundation of China(11271127 and 11061009)Science Research Program of Guizhou(GJ[2011]2367)the Co-Construction Project of Beijing Municipal Commission of Education
文摘In this paper, we extend the applications of proper orthogonal decomposition (POD) method, i.e., apply POD method to a mixed finite element (MFE) formulation naturally satisfied Brezz-Babu^ka for parabolic equations, establish a reduced-order MFE formulation with lower dimensions and sufficiently high accuracy, and provide the error estimates between the reduced-order POD MFE solutions and the classical MFE solutions and the implementation of algorithm for solving reduced-order MFE formulation. Some numerical examples illustrate the fact that the results of numerical computation are consis- tent with theoretical conclusions. Moreover, it is shown that the new reduced-order MFE formulation based on POD method is feasible and efficient for solving MFE formulation for parabolic equations.
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 Science Foundation of China(41275063 and 11401575)
文摘In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations (△-e/et)u(x,t)+q(x,t)u^p(x,t)=0 and nonlinear parabolic equations (△-e/et)u(x,t)+h(x,t)u^p(x,t)=0(p 〉 1) on Riemannian manifolds.As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang ([1], Bull. London Math. Soc. 38(2006), 1045-1053) and the author ([2], Nonlinear Anal. 74 (2011), 5141-5146).
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Partially supported by the project-sponsored by SRF for ROCS, SEM
文摘The author studies semilinear parabolic equations with initial and periodic boundary value conditions. In the presence of non-well-ordered sub- and super-solutions: "subsolution ≮ supersolution', the existence and stability/instability of equilibrium solutions are obtained.
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 presents alternating direction finite volume element methods for three-dimensional parabolic partial differential equations and gives four computational schemes, one is analogous to Douglas finite difference scheme with second-order splitting error, the other two schemes have third-order splitting error, and the last one is an extended LOD scheme. The L2 norm and H1 semi-norm error estimates are obtained for the first scheme and second one, respectively. Finally, two numerical examples are provided to illustrate the efficiency and accuracy of the methods.