This paper deals with the blow-up properties of the solution to a semilinear parabolic system with localized nonlinear reaction terms, subject to the null Dirichlet boundary condition. We first give sufficient conditi...This paper deals with the blow-up properties of the solution to a semilinear parabolic system with localized nonlinear reaction terms, subject to the null Dirichlet boundary condition. We first give sufficient conditions for that the classical solution blows up in the finite time, secondly give necessary conditions and a sufficient condition for that two components blow up simultaneously, and then obtain the uniform blow-up profiles in the interior. Finally we describe the asymptotic behavior of the blow-up solution in the boundary layer.展开更多
This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries. By using super-and sub-solution techniques, we first give the sufficient conditions that the classical solu...This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries. By using super-and sub-solution techniques, we first give the sufficient conditions that the classical solution exists globally and blows up in a finite time respectively, and then give the necessary and sufficient conditions that two components u and ν blow up simultaneously. Finally, the uniform blow-up profiles in the interior are presented.展开更多
The boundary value problem for the nonlinear parabolic system is solved by the finite difference method with nonuniform meshes. The existence and a priori estemates of the discrete vector solutions for the general dif...The boundary value problem for the nonlinear parabolic system is solved by the finite difference method with nonuniform meshes. The existence and a priori estemates of the discrete vector solutions for the general difference schemes with unequal meshsteps are established by the fixed point technique. The absolute and relative convergence of the discrete vector solution are justified by a series of a priori estimates. The analysis of mentioned problems are based on the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem of the nonlinear parabolic system.展开更多
For solving nonlinear parabolic equation on massive parallel computers, the construction of parallel difference schemes with simple design, high parallelism and unconditional stability and second order global accuracy...For solving nonlinear parabolic equation on massive parallel computers, the construction of parallel difference schemes with simple design, high parallelism and unconditional stability and second order global accuracy in space, has long been desired. In the present work, a new kind of general parallel difference schemes for the nonlinear parabolic system is proposed. The general parallel difference schemes include, among others, two new parallel schemes. In one of them, to obtain the interface values on the interface of sub-domains an explicit scheme of Jacobian type is employed, and then the fully implicit scheme is used in the sub-domains. Here, in the explicit scheme of Jacobian type, the values at the points being adjacent to the interface points are taken as the linear combination of values of previous two time layers at the adjoining points of the inner interface. For the construction of another new parallel difference scheme, the main procedure is as follows. Firstly the linear combination of values of previous two time layers at the interface points among the sub-domains is used as the （Dirichlet） boundary condition for solving the sub-domain problems. Then the values in the sub-domains are calculated by the fully implicit scheme. Finally the interface values are computed by the fully implicit scheme, and in fact these calculations of the last step are explicit since the values adjacent to the interface points have been obtained in the previous step. The existence, uniqueness, unconditional stability and the second order accuracy of the discrete vector solutions for the parallel difference schemes are proved. Numerical results are presented to examine the stability, accuracy and parallelism of the parallel schemes.展开更多
The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality(DEHVI)which couples an abstract parabolic evolution hemivariational inequality and a nonlinear dif...The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality(DEHVI)which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space.First,by apply ing surjectivity result for pseudomonotone multivalued mappins and the properties of Clarke's subgradient,we show the nonempty of the solution set for the parabolic hemivariational inequality.Then,some topological properties of the solution set are established such as boundedness,closedness and convexity.Furthermore,we explore the upper semicontinuity of the solution mapping.Finally,we prove the solution set of the system(DEHVI)is nonempty and the set of all trajectories of(DEHVI)is weakly compact in C(I,X).展开更多
In this paper, we study a nonlinear hyperbolic-parabolic system modeling some biological phenomena. By semigroup theory and Leray-Schauder fixed point argument, the local existence and uniqueness of the weak solutions...In this paper, we study a nonlinear hyperbolic-parabolic system modeling some biological phenomena. By semigroup theory and Leray-Schauder fixed point argument, the local existence and uniqueness of the weak solutions for this system are proved. For the spatial dimension N = 1, the global existence of the weak solution will be established by the bootstrap argument.展开更多
In this paper some new parallel difference schemes with interface extrapolation terms for a quasi-linear parabolic system of equations are constructed. Two types of time extrapolations are proposed to give the interfa...In this paper some new parallel difference schemes with interface extrapolation terms for a quasi-linear parabolic system of equations are constructed. Two types of time extrapolations are proposed to give the interface values on the interface of sub-domains or the values adjacent to the interface points, so that the unconditional stable parallel schemes with the second accuracy are formed. Without assuming heuristically that the original boundary value problem has the unique smooth vector solution, the existence and uniqueness of the discrete vector solutions of the parallel difference schemes constructed are proved. Moreover the unconditional stability of the parallel difference schemes is justified in the sense of the continuous dependence of the discrete vector solution of the schemes on the discrete known data of the original problems in the discrete W2(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the parallel difference schemes with interface extrapolation terms to the unique generalized solution of the original quasi-linear parabolic problem is proved. Numerical results are presented to show the good performance of the parallel schemes, including the unconditional stability, the second accuracy and the high parallelism.展开更多
The general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded measurable coefficients is studied. By the approach of...The general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded measurable coefficients is studied. By the approach of the discrete functional analysis, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general difference schemes with intrinsic parallelism justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete initial data of the original problems in the discrete W_2^(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original semilinear parabolic problem is proved.展开更多
The photovoltaic/thermal(PV/T)system is a promising option for countering energy shortages.To improve the performance of PV/T systems,compound parabolic concentrators(CPCs)and phase-change materials(PCMs)were jointly ...The photovoltaic/thermal(PV/T)system is a promising option for countering energy shortages.To improve the performance of PV/T systems,compound parabolic concentrators(CPCs)and phase-change materials(PCMs)were jointly applied to construct a concentrating photovoltaic/thermal system integrated with phase-change materials(PV/T-CPCM).An open-air environment is used to analyze the effects of different parameters and the intermittent operation strategy on the system performance.The results indicate that the short-circuit current and open-circuit voltage are positively correlated with the solar irradiance,but the open-circuit voltage is negatively correlated with the temperature of the PV modules.When the solar irradiance is 500 W⋅m^(−2) and the temperature of the PV modules is 27.5℃,the short-circuit current and open-circuit voltage are 1.0 A and 44.5 V,respectively.Higher solar irradiance results in higher thermal power,whereas the thermal efficiency is under lower solar irradiance(136.2-167.1 W⋅m^(−2) is twice under higher solar irradiance(272.3-455.7 W⋅m^(−2))).In addition,a higher mass flow rate corresponds to a better cooling effect and greater pump energy consumption.When the mass flow rate increases from 0.01 to 0.02 kg⋅s^(-1),the temperature difference between the inlet and outlet decreases by 1.8℃,and the primary energy-saving efficiency decreases by 0.53%.The intermittent operation of a water pump can reduce the energy consumption of the system,and the combination of liquid cooling with PCMs provides better thermal regulation and energy-saving effects under various conditions.展开更多
This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut - △u - αut =vp, vt -△v - α△vt = uq with p, q≥ 1 and pq 〉 1, where the viscous terms of third order are included. We first fin...This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut - △u - αut =vp, vt -△v - α△vt = uq with p, q≥ 1 and pq 〉 1, where the viscous terms of third order are included. We first find the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. Moreover, time-decay profiles are obtained for the global solutions. It can be found that, different from those for the situations of general semilinear heat systems, we have to use distinctive techniques to treat the influence from the viscous terms of the highest order. To fix the non-global solutions, we exploit the test function method, instead of the general Kaplan method for heat systems. To obtain the global solutions, we apply the LP-Lq technique to establish some uniform Lm time-decay estimates. In particular, under a suitable classification for the nonlinear parameters and the initial data, various Lm time-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system. It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing effect to establish the compactness of approximating solutions, which cannot be directly realized here due to absence of the smooth effect in the pseudo-parabolic system.展开更多
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 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
文摘This paper deals with the blow-up properties of the solution to a semilinear parabolic system with localized nonlinear reaction terms, subject to the null Dirichlet boundary condition. We first give sufficient conditions for that the classical solution blows up in the finite time, secondly give necessary conditions and a sufficient condition for that two components blow up simultaneously, and then obtain the uniform blow-up profiles in the interior. Finally we describe the asymptotic behavior of the blow-up solution in the boundary layer.
基金This work was supported by the National Natural Science Foundation of China (Grant Nos.10471013,10471022)the Ministry of Education of China Science and Technology Major Projects (Grant No.104090)
文摘This paper deals with a semi-linear parabolic system with nonlinear nonlocal sources and nonlocal boundaries. By using super-and sub-solution techniques, we first give the sufficient conditions that the classical solution exists globally and blows up in a finite time respectively, and then give the necessary and sufficient conditions that two components u and ν blow up simultaneously. Finally, the uniform blow-up profiles in the interior are presented.
文摘The boundary value problem for the nonlinear parabolic system is solved by the finite difference method with nonuniform meshes. The existence and a priori estemates of the discrete vector solutions for the general difference schemes with unequal meshsteps are established by the fixed point technique. The absolute and relative convergence of the discrete vector solution are justified by a series of a priori estimates. The analysis of mentioned problems are based on the assumption of heuristic character concerning the existence of the unique smooth solution for the original problem of the nonlinear parabolic system.
基金The project is supported by the Special Funds for Major State Basic Research Projects 2005CB321703, the National Nature Science Foundation of China （No. 10476002, 60533020）.
文摘For solving nonlinear parabolic equation on massive parallel computers, the construction of parallel difference schemes with simple design, high parallelism and unconditional stability and second order global accuracy in space, has long been desired. In the present work, a new kind of general parallel difference schemes for the nonlinear parabolic system is proposed. The general parallel difference schemes include, among others, two new parallel schemes. In one of them, to obtain the interface values on the interface of sub-domains an explicit scheme of Jacobian type is employed, and then the fully implicit scheme is used in the sub-domains. Here, in the explicit scheme of Jacobian type, the values at the points being adjacent to the interface points are taken as the linear combination of values of previous two time layers at the adjoining points of the inner interface. For the construction of another new parallel difference scheme, the main procedure is as follows. Firstly the linear combination of values of previous two time layers at the interface points among the sub-domains is used as the （Dirichlet） boundary condition for solving the sub-domain problems. Then the values in the sub-domains are calculated by the fully implicit scheme. Finally the interface values are computed by the fully implicit scheme, and in fact these calculations of the last step are explicit since the values adjacent to the interface points have been obtained in the previous step. The existence, uniqueness, unconditional stability and the second order accuracy of the discrete vector solutions for the parallel difference schemes are proved. Numerical results are presented to examine the stability, accuracy and parallelism of the parallel schemes.
基金NSF of Guangxi(Grant No.2023GXNSFAA026085)Guangxi Science and Technology Department Specific Research Project of Guangxi for Research Bases and Talents(Grant No.AD23023001)+1 种基金NNSF of China Grant Nos.12071413,12111530282 the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement No.823731 CONMECHthe Innovation Project of Guangxi University for Nationalities(Grant No.gxun-chxps202072)。
文摘The goal of this paper is to deal with a new dynamic system called a differential evolution hemivariational inequality(DEHVI)which couples an abstract parabolic evolution hemivariational inequality and a nonlinear differential equation in a Banach space.First,by apply ing surjectivity result for pseudomonotone multivalued mappins and the properties of Clarke's subgradient,we show the nonempty of the solution set for the parabolic hemivariational inequality.Then,some topological properties of the solution set are established such as boundedness,closedness and convexity.Furthermore,we explore the upper semicontinuity of the solution mapping.Finally,we prove the solution set of the system(DEHVI)is nonempty and the set of all trajectories of(DEHVI)is weakly compact in C(I,X).
文摘In this paper, we study a nonlinear hyperbolic-parabolic system modeling some biological phenomena. By semigroup theory and Leray-Schauder fixed point argument, the local existence and uniqueness of the weak solutions for this system are proved. For the spatial dimension N = 1, the global existence of the weak solution will be established by the bootstrap argument.
基金This work was supported by the Special Funds for Major State Basic Research Projects (Grant No.2005CB321703)the National Natural Science Foundation of China (Grant Nos. 10476002, 60533020)the Science Foundation of CAEP (Grant No. 20060649)
文摘In this paper some new parallel difference schemes with interface extrapolation terms for a quasi-linear parabolic system of equations are constructed. Two types of time extrapolations are proposed to give the interface values on the interface of sub-domains or the values adjacent to the interface points, so that the unconditional stable parallel schemes with the second accuracy are formed. Without assuming heuristically that the original boundary value problem has the unique smooth vector solution, the existence and uniqueness of the discrete vector solutions of the parallel difference schemes constructed are proved. Moreover the unconditional stability of the parallel difference schemes is justified in the sense of the continuous dependence of the discrete vector solution of the schemes on the discrete known data of the original problems in the discrete W2(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the parallel difference schemes with interface extrapolation terms to the unique generalized solution of the original quasi-linear parabolic problem is proved. Numerical results are presented to show the good performance of the parallel schemes, including the unconditional stability, the second accuracy and the high parallelism.
基金Project supported by China "National Key Program for Developing Basic Sciences" (No.G1999032801) the National Natural Science Foundation of China (No.19932010) the Science and Technology Foundation of Chinese Academy of Engineering Physics (No.200206
文摘The general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded measurable coefficients is studied. By the approach of the discrete functional analysis, the existence and uniqueness of the discrete vector solutions of the nonlinear difference system with intrinsic parallelism are proved. Moreover the unconditional stability of the general difference schemes with intrinsic parallelism justified in the sense of the continuous dependence of the discrete vector solution of the difference schemes on the discrete initial data of the original problems in the discrete W_2^(2,1) (Q△) norms. Finally the convergence of the discrete vector solutions of the certain difference schemes with intrinsic parallelism to the unique generalized solution of the original semilinear parabolic problem is proved.
基金supported by the Hebei Province Postdoctoral Merit Funding Program(Grant No.:B2022005004)the Science and Tech-nology Nova Plan of Hebei University of Technology(Grant No.:JBKYXX2207)+2 种基金the National Natural Science Foundation of China(Grant No.:51978231)the S&T Program of Hebei(Project No.:216Z4502G)the Natural Science Foundation of Hebei Province(Grant No.:E2020202196).
文摘The photovoltaic/thermal(PV/T)system is a promising option for countering energy shortages.To improve the performance of PV/T systems,compound parabolic concentrators(CPCs)and phase-change materials(PCMs)were jointly applied to construct a concentrating photovoltaic/thermal system integrated with phase-change materials(PV/T-CPCM).An open-air environment is used to analyze the effects of different parameters and the intermittent operation strategy on the system performance.The results indicate that the short-circuit current and open-circuit voltage are positively correlated with the solar irradiance,but the open-circuit voltage is negatively correlated with the temperature of the PV modules.When the solar irradiance is 500 W⋅m^(−2) and the temperature of the PV modules is 27.5℃,the short-circuit current and open-circuit voltage are 1.0 A and 44.5 V,respectively.Higher solar irradiance results in higher thermal power,whereas the thermal efficiency is under lower solar irradiance(136.2-167.1 W⋅m^(−2) is twice under higher solar irradiance(272.3-455.7 W⋅m^(−2))).In addition,a higher mass flow rate corresponds to a better cooling effect and greater pump energy consumption.When the mass flow rate increases from 0.01 to 0.02 kg⋅s^(-1),the temperature difference between the inlet and outlet decreases by 1.8℃,and the primary energy-saving efficiency decreases by 0.53%.The intermittent operation of a water pump can reduce the energy consumption of the system,and the combination of liquid cooling with PCMs provides better thermal regulation and energy-saving effects under various conditions.
基金supported by National Natural Science Foundation of China(Grant Nos.11171048 and 11201047)the Doctor Startup Foundation of Liaoning Province(Grant No.20121025)the Fundamental Research Funds for the Central Universities
文摘This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut - △u - αut =vp, vt -△v - α△vt = uq with p, q≥ 1 and pq 〉 1, where the viscous terms of third order are included. We first find the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. Moreover, time-decay profiles are obtained for the global solutions. It can be found that, different from those for the situations of general semilinear heat systems, we have to use distinctive techniques to treat the influence from the viscous terms of the highest order. To fix the non-global solutions, we exploit the test function method, instead of the general Kaplan method for heat systems. To obtain the global solutions, we apply the LP-Lq technique to establish some uniform Lm time-decay estimates. In particular, under a suitable classification for the nonlinear parameters and the initial data, various Lm time-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system. It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing effect to establish the compactness of approximating solutions, which cannot be directly realized here due to absence of the smooth effect in the pseudo-parabolic system.
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).
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.
