In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of ...In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of degree m, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133).展开更多
This paper deals with positive solutions to a class of nonlocal and degenerate quasilinear parabolic system with null Dirichlet boundary conditions. The blow-up rate and blow-up profile are gained if the parameters an...This paper deals with positive solutions to a class of nonlocal and degenerate quasilinear parabolic system with null Dirichlet boundary conditions. The blow-up rate and blow-up profile are gained if the parameters and the initial data satisfy some conditions.展开更多
This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fuji...This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel'dovich-Kompaneetz-Barenblatt profile展开更多
In this paper, we consider the Cauchy problem of a class of semilinear parabolic system. Firstly, we obtain the local existence of solutions of (1.1),(1.2) in H-1(R(1)), and then we prove the global existence of the s...In this paper, we consider the Cauchy problem of a class of semilinear parabolic system. Firstly, we obtain the local existence of solutions of (1.1),(1.2) in H-1(R(1)), and then we prove the global existence of the solutions in H-1 through a priori estimate.展开更多
We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonli...We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form { b1(x,u1)/ t-div(a(x,t,u1,Du1))+div(Ф1(u1))+f1(x,u1,u2)=O in Q, b2(x,u2)/ t-div(a(x,t,u2,Du2))+div(Ф2(u2))+f2(x,u1,u2)=O in Q in the framework of weighted Sobolev spaces, where b(x,u) is unbounded function on u, the Carath6odory function ai satisfying the coercivity condition, the general growth condition and only the large monotonicity, the function Фi is assumed to be continuous on ]R and not belong to (Lloc1(Q))N.展开更多
In this paper, we study a nonlinear parabolic system with variable exponents. The existence of classical solutions to an initial and boundary value problem is obtained by a fixed point theorem of the contraction mappi...In this paper, we study a nonlinear parabolic system with variable exponents. The existence of classical solutions to an initial and boundary value problem is obtained by a fixed point theorem of the contraction mapping, and the blow-up property of solutions in finite time is obtained with the help of the eigenfunction of the Laplace equation and a delicated estimate.展开更多
The existence of a solution to the parabolic system with the fractional Laplacian (-△) α/2, α 〉 0 is proven, this solution decays at different rates along different time sequences going to infinity. As an applic...The existence of a solution to the parabolic system with the fractional Laplacian (-△) α/2, α 〉 0 is proven, this solution decays at different rates along different time sequences going to infinity. As an application, the existence of a solution to the generalized Navier-Stokes equations is proven, which decays at different rates along different time sequences going to infinity. The generalized Navier-Stokes equations are the equations resulting from replacing -△ in the Navier-Stokes equations by (-△)^m, m〉 0. At last, a similar result for 3-D incompressible anisotropic Navier-Stokes system is obtained.展开更多
This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. ...This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.展开更多
In this paper, the existence and uniqueness of local solutions to the initial and boundary value problem of a class of parabolic system related to the p-Laplacian are studied. The regularization method is used to cons...In this paper, the existence and uniqueness of local solutions to the initial and boundary value problem of a class of parabolic system related to the p-Laplacian are studied. The regularization method is used to construct a sequence of approximation solutions, with the help of monotone iteration technique, then we get the existence of solution of a regularized system. By the use of a standard limiting process, the existence of the local solutions of the system is obtained. Finally, the uniqueness of the solution is also proven.展开更多
This paper deals with the blow-up properties of solutions to the systems ut u(t) - Delta u = e(v(xo,t)), v(t) - Delta v = e(u(xo,t)) in Omega x (0,T) subject to either initial conditions or the initial and boundary-va...This paper deals with the blow-up properties of solutions to the systems ut u(t) - Delta u = e(v(xo,t)), v(t) - Delta v = e(u(xo,t)) in Omega x (0,T) subject to either initial conditions or the initial and boundary-value conditions. The authors show that under certain conditions the solution blows up in finite time and prove that the set of all blow-up points is the whole region. Moreover, the exact blow-up rates are also derived.展开更多
Blow-up phenomena for solutions of some nonlinear parabolic systems with time dependent coefficients are investigated. Both lower and upper bounds for the blow-up time are derived when blow-up occurs.
A nonlinear parabolic system is derived to describe incompressible nuclear waste-disposal contamination in porous media. A sequential implicit tirne-stepping is defined, in which the pressure and Darcy velocity of the...A nonlinear parabolic system is derived to describe incompressible nuclear waste-disposal contamination in porous media. A sequential implicit tirne-stepping is defined, in which the pressure and Darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the brine, radionuclid and heat are treated by a combination of a Galerkin finite element method and the method of characteristics. Optimal-order convergence in L2 is proved. Time-truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the brine, radionuclide and heal equalios, temporal and spatial error are lossened by direct compulation of the velocity in the mixed method, as opposed to differentiation of the pressure.展开更多
In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type u1=-A(x, t)ux4+B(x, t)ux2+(g(u))x2+(grad h(u))x+f(u)are studied, where u(x, t)=(u1...In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type u1=-A(x, t)ux4+B(x, t)ux2+(g(u))x2+(grad h(u))x+f(u)are studied, where u(x, t)=(u1(x, t).…, uJ(x, t) is a J-dimensional unknown vector valued function, f(u) and g(u) are the J-dimensional vector valued function of u(x, t), h(u) is a scalar function of u, A(x, t) and B(x, t) are J×J matrices of functions. The existent, uniqueness and regularities of the generalized global solution and classical global solution of the problems are proved. When J=1, h(u)=0, g(u)=au3, A=a1, B=a2, where a1, a2 a are constants, the system is a generalized diffusion model equation in population problem.展开更多
In this paper, time-optimal control problem for a liner n× n co-operative parabolic system involving Laplace operator is considered. This problem is, steering an initial state y(0)=u?, with control u?so that an o...In this paper, time-optimal control problem for a liner n× n co-operative parabolic system involving Laplace operator is considered. This problem is, steering an initial state y(0)=u?, with control u?so that an observation y(t) hitting a given target set in minimum time. First, the existence and uniqueness of solutions of such system under conditions on the coefficients are proved. Afterwards necessary and sufficient conditions of optimality are obtained. Finally a scaler case is given.展开更多
The problems of optimal control (OCPs) related to PDEs are a very active area of research. These problems deal with the processes of mechanical engineering, heat aeronautics, physics, hydro and gas dynamics, the physi...The problems of optimal control (OCPs) related to PDEs are a very active area of research. These problems deal with the processes of mechanical engineering, heat aeronautics, physics, hydro and gas dynamics, the physics of plasma and other real life problems. In this paper, we deal with a class of the constrained OCP for parabolic systems. It is converted to new unconstrained OCP by adding a penalty function to the cost functional. The existence solution of the considering system of parabolic optimal control problem (POCP) is introduced. In this way, the uniqueness theorem for the solving POCP is introduced. Therefore, a theorem for the sufficient differentiability conditions has been proved.展开更多
This paper studies quenching properties of solutions of a semilinear parabolic system with localized reaction sources in a square domain. The system has the homogeneous Dirichlet boundary condition and null initial co...This paper studies quenching properties of solutions of a semilinear parabolic system with localized reaction sources in a square domain. The system has the homogeneous Dirichlet boundary condition and null initial condition. We prove that solutions quench simultaneously, and compute approximated critical values of the system using a numerical method.展开更多
In this paper, we shall study the stabilization and the robustness of a constrained feedback control for bilinear parabolic systems defined on a Hilbert state space. Then, we shall show that stabilizing such a system ...In this paper, we shall study the stabilization and the robustness of a constrained feedback control for bilinear parabolic systems defined on a Hilbert state space. Then, we shall show that stabilizing such a system reduces stabilization only in its projection on a suitable subspace. For this purpose, a new constrained stabilizing feedback control that allows a polynomial decay estimate of the stabilized state is given. Also, the robustness of the considered control is discussed. An illustrating example and simulations are presented.展开更多
The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achie...The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achieving internal and boundary regional controllability. The approach is based on an extension of the Hilbert Uniqueness Method (HUM) and Schauder’s fixed point theorem. We give a numerical example developed in internal and boundary sub region. These numerical illustrations show the efficiency of the approach and lead to conjectures.展开更多
This paper is focused on studying an important concept of the system analysis, which is the regional enlarged observability or constrained observability of the gradient for distributed parabolic systems evolving in th...This paper is focused on studying an important concept of the system analysis, which is the regional enlarged observability or constrained observability of the gradient for distributed parabolic systems evolving in the spatial domain Ω We will explore an approach based on the Hilbert Uniqueness Method (HUM), which can reconstruct the initial gradient state between two prescribed functions f1 and f2 only in a critical subregion ω of Ω without the knowledge of the state. Finally, the obtained results are illustrated by numerical simulations.展开更多
A parabolic trough solar collector(PTSC)converts solar radiation into thermal energy.However,low thermal efficiency of PTSC poses a hindrance to the deployment of solar thermal power plants.Thermal performance of PTSC...A parabolic trough solar collector(PTSC)converts solar radiation into thermal energy.However,low thermal efficiency of PTSC poses a hindrance to the deployment of solar thermal power plants.Thermal performance of PTSC is enhanced in this study by incorporating magnetic nanoparticles into the working fluid.The circular receiver pipe,with dimensions of 66 mm diameter,2 mm thickness,and 24 m length,is exposed to uniform temperature and velocity conditions.The working fluid,Therminol-66,is supplemented with Fe3O4 magnetic nanoparticles at concentrations ranging from 1%to 4%.The findings demonstrate that the inclusion of nanoparticles increases the convective heat transfer coefficient(HTC)of the PTSC,with higher nanoparticle volume fractions leading to greater heat transfer but increased pressure drop.The thermal enhancement factor(TEF)of the PTSC is positively affected by the volume fraction of nanoparticles,both with and without a magnetic field.Notably,the scenario with a 4%nanoparticle volume fraction and a magnetic field strength of 250 G exhibits the highest TEF,indicating superior thermal performance.These findings offer potential avenues for improving the efficiency of PTSCs in solar thermal plants by introducing magnetic nanoparticles into the working fluid.展开更多
基金supported by the Zhejiang Provincial Natural Science Foundation of China(LY21A010016)the National Natural Science Foundation of China(11901550).
文摘In this paper, we investigate a blow-up phenomenon for a semilinear parabolic system on locally finite graphs. Under some appropriate assumptions on the curvature condition CDE’(n,0), the polynomial volume growth of degree m, the initial values, and the exponents in absorption terms, we prove that every non-negative solution of the semilinear parabolic system blows up in a finite time. Our current work extends the results achieved by Lin and Wu (Calc Var Partial Differ Equ, 2017, 56: Art 102) and Wu (Rev R Acad Cien Serie A Mat, 2021, 115: Art 133).
基金supported by the National Natural Science Foundation of China (No.10671210)the Foundation of Jiangsu Education Commission (No.07KJD110166)+1 种基金the Postdoctoral Research Foundation of Jiangsu Province (No.0702004C)the Project of Nantong University (Nos.06Z011,08B02)
文摘This paper deals with positive solutions to a class of nonlocal and degenerate quasilinear parabolic system with null Dirichlet boundary conditions. The blow-up rate and blow-up profile are gained if the parameters and the initial data satisfy some conditions.
基金supported in part by NSF of China (11071266)in part by NSF project of CQ CSTC (2010BB9218)partially supported by the Educational Science Foundation of Chongqing(KJ101303) China
文摘This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel'dovich-Kompaneetz-Barenblatt profile
文摘In this paper, we consider the Cauchy problem of a class of semilinear parabolic system. Firstly, we obtain the local existence of solutions of (1.1),(1.2) in H-1(R(1)), and then we prove the global existence of the solutions in H-1 through a priori estimate.
文摘We prove an existence result without assumptions on the growth of some nonlinear terms, and the existence of a renormalized solution. In this work, we study the existence of renormalized solutions for a class of nonlinear parabolic systems with three unbounded nonlinearities, in the form { b1(x,u1)/ t-div(a(x,t,u1,Du1))+div(Ф1(u1))+f1(x,u1,u2)=O in Q, b2(x,u2)/ t-div(a(x,t,u2,Du2))+div(Ф2(u2))+f2(x,u1,u2)=O in Q in the framework of weighted Sobolev spaces, where b(x,u) is unbounded function on u, the Carath6odory function ai satisfying the coercivity condition, the general growth condition and only the large monotonicity, the function Фi is assumed to be continuous on ]R and not belong to (Lloc1(Q))N.
文摘In this paper, we study a nonlinear parabolic system with variable exponents. The existence of classical solutions to an initial and boundary value problem is obtained by a fixed point theorem of the contraction mapping, and the blow-up property of solutions in finite time is obtained with the help of the eigenfunction of the Laplace equation and a delicated estimate.
基金Supported by the National Natural Science Foundation of China (1057115810871175)
文摘The existence of a solution to the parabolic system with the fractional Laplacian (-△) α/2, α 〉 0 is proven, this solution decays at different rates along different time sequences going to infinity. As an application, the existence of a solution to the generalized Navier-Stokes equations is proven, which decays at different rates along different time sequences going to infinity. The generalized Navier-Stokes equations are the equations resulting from replacing -△ in the Navier-Stokes equations by (-△)^m, m〉 0. At last, a similar result for 3-D incompressible anisotropic Navier-Stokes system is obtained.
文摘This paper deals with some initial-oblique derivative boundary value problems for nonlinear nondivergent parabolic systems of several second order equations with measurable coefficients in multiply connected domains. Firstly, a priori estimates of solutions for the initial-boundary value problems are given, and then by using the above estimates of solutions and the Leray-Schauder theorem, the existence and uniqueness of solutions for the problems are proved.
文摘In this paper, the existence and uniqueness of local solutions to the initial and boundary value problem of a class of parabolic system related to the p-Laplacian are studied. The regularization method is used to construct a sequence of approximation solutions, with the help of monotone iteration technique, then we get the existence of solution of a regularized system. By the use of a standard limiting process, the existence of the local solutions of the system is obtained. Finally, the uniqueness of the solution is also proven.
文摘This paper deals with the blow-up properties of solutions to the systems ut u(t) - Delta u = e(v(xo,t)), v(t) - Delta v = e(u(xo,t)) in Omega x (0,T) subject to either initial conditions or the initial and boundary-value conditions. The authors show that under certain conditions the solution blows up in finite time and prove that the set of all blow-up points is the whole region. Moreover, the exact blow-up rates are also derived.
文摘Blow-up phenomena for solutions of some nonlinear parabolic systems with time dependent coefficients are investigated. Both lower and upper bounds for the blow-up time are derived when blow-up occurs.
基金The research was supported by the Natural Science Foundation of China
文摘A nonlinear parabolic system is derived to describe incompressible nuclear waste-disposal contamination in porous media. A sequential implicit tirne-stepping is defined, in which the pressure and Darcy velocity of the mixture are approximated simultaneously by a mixed finite element method and the brine, radionuclid and heat are treated by a combination of a Galerkin finite element method and the method of characteristics. Optimal-order convergence in L2 is proved. Time-truncation errors of standard procedures are reduced by time stepping along the characteristics of the hyperbolic part of the brine, radionuclide and heal equalios, temporal and spatial error are lossened by direct compulation of the velocity in the mixed method, as opposed to differentiation of the pressure.
文摘In this paper, the periodic boundary problem and the initial value problem for the nonlinear system of parabolic type u1=-A(x, t)ux4+B(x, t)ux2+(g(u))x2+(grad h(u))x+f(u)are studied, where u(x, t)=(u1(x, t).…, uJ(x, t) is a J-dimensional unknown vector valued function, f(u) and g(u) are the J-dimensional vector valued function of u(x, t), h(u) is a scalar function of u, A(x, t) and B(x, t) are J×J matrices of functions. The existent, uniqueness and regularities of the generalized global solution and classical global solution of the problems are proved. When J=1, h(u)=0, g(u)=au3, A=a1, B=a2, where a1, a2 a are constants, the system is a generalized diffusion model equation in population problem.
文摘In this paper, time-optimal control problem for a liner n× n co-operative parabolic system involving Laplace operator is considered. This problem is, steering an initial state y(0)=u?, with control u?so that an observation y(t) hitting a given target set in minimum time. First, the existence and uniqueness of solutions of such system under conditions on the coefficients are proved. Afterwards necessary and sufficient conditions of optimality are obtained. Finally a scaler case is given.
文摘The problems of optimal control (OCPs) related to PDEs are a very active area of research. These problems deal with the processes of mechanical engineering, heat aeronautics, physics, hydro and gas dynamics, the physics of plasma and other real life problems. In this paper, we deal with a class of the constrained OCP for parabolic systems. It is converted to new unconstrained OCP by adding a penalty function to the cost functional. The existence solution of the considering system of parabolic optimal control problem (POCP) is introduced. In this way, the uniqueness theorem for the solving POCP is introduced. Therefore, a theorem for the sufficient differentiability conditions has been proved.
文摘This paper studies quenching properties of solutions of a semilinear parabolic system with localized reaction sources in a square domain. The system has the homogeneous Dirichlet boundary condition and null initial condition. We prove that solutions quench simultaneously, and compute approximated critical values of the system using a numerical method.
文摘In this paper, we shall study the stabilization and the robustness of a constrained feedback control for bilinear parabolic systems defined on a Hilbert state space. Then, we shall show that stabilizing such a system reduces stabilization only in its projection on a suitable subspace. For this purpose, a new constrained stabilizing feedback control that allows a polynomial decay estimate of the stabilized state is given. Also, the robustness of the considered control is discussed. An illustrating example and simulations are presented.
文摘The aim of this brief paper is to give several results concerning the regional controllability of distributed systems governed by semi-linear parabolic equations. We concentrate on the determination of a control achieving internal and boundary regional controllability. The approach is based on an extension of the Hilbert Uniqueness Method (HUM) and Schauder’s fixed point theorem. We give a numerical example developed in internal and boundary sub region. These numerical illustrations show the efficiency of the approach and lead to conjectures.
文摘This paper is focused on studying an important concept of the system analysis, which is the regional enlarged observability or constrained observability of the gradient for distributed parabolic systems evolving in the spatial domain Ω We will explore an approach based on the Hilbert Uniqueness Method (HUM), which can reconstruct the initial gradient state between two prescribed functions f1 and f2 only in a critical subregion ω of Ω without the knowledge of the state. Finally, the obtained results are illustrated by numerical simulations.
文摘A parabolic trough solar collector(PTSC)converts solar radiation into thermal energy.However,low thermal efficiency of PTSC poses a hindrance to the deployment of solar thermal power plants.Thermal performance of PTSC is enhanced in this study by incorporating magnetic nanoparticles into the working fluid.The circular receiver pipe,with dimensions of 66 mm diameter,2 mm thickness,and 24 m length,is exposed to uniform temperature and velocity conditions.The working fluid,Therminol-66,is supplemented with Fe3O4 magnetic nanoparticles at concentrations ranging from 1%to 4%.The findings demonstrate that the inclusion of nanoparticles increases the convective heat transfer coefficient(HTC)of the PTSC,with higher nanoparticle volume fractions leading to greater heat transfer but increased pressure drop.The thermal enhancement factor(TEF)of the PTSC is positively affected by the volume fraction of nanoparticles,both with and without a magnetic field.Notably,the scenario with a 4%nanoparticle volume fraction and a magnetic field strength of 250 G exhibits the highest TEF,indicating superior thermal performance.These findings offer potential avenues for improving the efficiency of PTSCs in solar thermal plants by introducing magnetic nanoparticles into the working fluid.