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.
This paper deals with propagations of singularities in solutions to a parabolic system coupled with nonlocal nonlinear sources. The estimates for the four possible blow-up rates as well as the boundary layer profiles ...This paper deals with propagations of singularities in solutions to a parabolic system coupled with nonlocal nonlinear sources. The estimates for the four possible blow-up rates as well as the boundary layer profiles are established. The critical exponent of the system is determined also.展开更多
In this note, the long-time behavior of solutions for the following parabolic systemsare studied, where Ω is a bounded smooth domain, v is the outer normal vector on Ω.u<sub>n</sub>,.B<sub>n</su...In this note, the long-time behavior of solutions for the following parabolic systemsare studied, where Ω is a bounded smooth domain, v is the outer normal vector on Ω.u<sub>n</sub>,.B<sub>n</sub>∈C<sup>1</sup>（Ω） and satisfy u<sub>0</sub>/v<sub>0</sub> = u<sub>0</sub><sup>m</sup>v<sub>0</sub><sup>n</sup>, v<sub>0</sub>/v = u<sub>0</sub><sup>p</sup>v<sub>0</sub><sup>q</sup>, x∈Ω;and u<sub>0</sub>（x）】0, v<sub>0</sub>（x）】0, x∈Ω.m,n,p. q are nonnegative constants. Our main result reads as follows.Theorem. Solutions of （1） exist globally展开更多
§1. Introduction In this paper we consider the parabolic system (?)t/(?)ui-div(|▽u|p-2▽ui)=0(1≤i≤m), with p>1, where ui=ui(x, t), ▽=gradxand x varies in an open domain Ω(?)RN. The system is degenerate if...§1. Introduction In this paper we consider the parabolic system (?)t/(?)ui-div(|▽u|p-2▽ui)=0(1≤i≤m), with p>1, where ui=ui(x, t), ▽=gradxand x varies in an open domain Ω(?)RN. The system is degenerate if p>2 or singular if 1<p<2. A vector function u=(u1, u2, …, um) defined in ΩT=Ω×[0, T] is called a solution of the system (1.1) if展开更多
This paper deals with the blow-up rate estimates of solutions for semilinear parabolic systems coupled in an equation and a boundary condition. The upper and lower bounds of blow-up rates have been obtained.
Under both controllable and natural growth conditions, the spatial derivatives D_auof a solution u∈ L^2(0, T; H^1(Q , R^N)) ∩L~∞(0, T; L^2(Q , R^N)) (or∩L~∞(Q , R^N)) of a nonlinearparabolic system u_t^i - D_αA...Under both controllable and natural growth conditions, the spatial derivatives D_auof a solution u∈ L^2(0, T; H^1(Q , R^N)) ∩L~∞(0, T; L^2(Q , R^N)) (or∩L~∞(Q , R^N)) of a nonlinearparabolic system u_t^i - D_αA_i~α(x, t, u, Du) = B_i(x, t, u, Du), i = 1,…, N, (x, t) ∈Q,in fact, belong to L_(loc)~p(Q,R^N) for some p>2. Every solution of a quasilinear parabolic system u_t^i - D_α[A_(ij)^(αβ)(x, t, u)D_βu^j +α_i~α(x, t, u)] = B_i(x, t, u, Du), i = 1, …, Nis Holder continuous in an open set Q_1?Q with H^(n+2-p)(Q\Q_1)=0. If A_(ij)^(αβ)(x,t,u)=0 when j>i, and the growth of B_i(x, t, u, p) w. r. t. |p| is less than 2, then Q_1=Q. If A_(ij)^(αβ) and α_i~αare Holder continuous, then so are D_αu in Q_1.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
The two-dimensional（2D） compound parabolic concentrator＇s（CPC） characteristics are analyzed.It is shown that CPC＇s height is taller and its light collecting ability is stronger with the CPC＇s field of view decr...The two-dimensional（2D） compound parabolic concentrator＇s（CPC） characteristics are analyzed.It is shown that CPC＇s height is taller and its light collecting ability is stronger with the CPC＇s field of view decreasing when the bottom radius is unchanged.According to the ZEMAX analysis,CPC is good at collecting optical signal,and the antenna combining CPC with hemispherical lens can gather more optical signal than a single CPC or CPCs combined in series.The light propagation of scattering optical communication based on multiple scattering is simulated by Monte Carlo method,and the results show that using CPC as receiving antenna can strengthen communication system＇s signal collecting ability and increase its communication distance.展开更多
This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation.Taking the finite dement pair Q11/Q01×Q10 as an example, a new mixed finite element method (FEM)i...This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation.Taking the finite dement pair Q11/Q01×Q10 as an example, a new mixed finite element method (FEM)is established and the r-independent superclose results of the original variable u in Hi-norm and the flux variable q=-a(u)■u in L^2- norm are deduced (τ is the temporal partition parameter).A key to our analysis is all error splitting technique,with which the time-discrete and the spatial-discrete systems are constructed,respectively.For the first system,tile boundedness of the temporal errors are obtained.For the second system,the spatial superclose results are presented unconditionally.while the previous literature always only obtain the convergent estimates or require certain time step conditions.Finally,some numerical results are provided to confirm the theoretical analysis,and show the efficiency of the proposed method.展开更多
In 1988, Yu . A. Alkhutov and I. T. Mamedov discussed the solvability of the Dirichlet problem for linear uniformly parabolic equations with measurable coefficients where the coefficients satisfy the condition In this...In 1988, Yu . A. Alkhutov and I. T. Mamedov discussed the solvability of the Dirichlet problem for linear uniformly parabolic equations with measurable coefficients where the coefficients satisfy the condition In this paper, we try to generalize the results of Alkhutov and Mamedov to nonlinear uni- formly parabolic systems of second order equations with measurable coefficients; moreover, we also discuss the solvability of the Neumann problem for the above systems.展开更多
This paper studies the problem of adaptive neural networks control(ANNC) for uncertain parabolic distributed parameter systems(DPSs) with nonlinear periodic time-varying parameter(NPTVP). Firstly, the uncertain nonlin...This paper studies the problem of adaptive neural networks control(ANNC) for uncertain parabolic distributed parameter systems(DPSs) with nonlinear periodic time-varying parameter(NPTVP). Firstly, the uncertain nonlinear dynamic and unknown periodic TVP are represented by using neural networks(NNs) and Fourier series expansion(FSE), respectively. Secondly, based on the ANNC and reparameterization approaches, two control algorithms are designed to make the uncertain parabolic DPSs with NPTVP asymptotically stable. The sufficient conditions of the asymptotically stable for the resulting closed-loop systems are also derived. Finally, a simulation is carried out to verify the effectiveness of the two control algorithms designed in this work.展开更多
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supported by the National Natural Science Foundation of China (Grant No. 10771024)
文摘This paper deals with propagations of singularities in solutions to a parabolic system coupled with nonlocal nonlinear sources. The estimates for the four possible blow-up rates as well as the boundary layer profiles are established. The critical exponent of the system is determined also.
基金Project supported by the National Natural Science Foundation of China.
文摘In this note, the long-time behavior of solutions for the following parabolic systemsare studied, where Ω is a bounded smooth domain, v is the outer normal vector on Ω.u<sub>n</sub>,.B<sub>n</sub>∈C<sup>1</sup>（Ω） and satisfy u<sub>0</sub>/v<sub>0</sub> = u<sub>0</sub><sup>m</sup>v<sub>0</sub><sup>n</sup>, v<sub>0</sub>/v = u<sub>0</sub><sup>p</sup>v<sub>0</sub><sup>q</sup>, x∈Ω;and u<sub>0</sub>（x）】0, v<sub>0</sub>（x）】0, x∈Ω.m,n,p. q are nonnegative constants. Our main result reads as follows.Theorem. Solutions of （1） exist globally
文摘§1. Introduction In this paper we consider the parabolic system (?)t/(?)ui-div(|▽u|p-2▽ui)=0(1≤i≤m), with p>1, where ui=ui(x, t), ▽=gradxand x varies in an open domain Ω(?)RN. The system is degenerate if p>2 or singular if 1<p<2. A vector function u=(u1, u2, …, um) defined in ΩT=Ω×[0, T] is called a solution of the system (1.1) if
基金the National Natural Science Foundation of China (Grant No. 19831060) Hwa Ying Culture & Education Foundation.
文摘This paper deals with the blow-up rate estimates of solutions for semilinear parabolic systems coupled in an equation and a boundary condition. The upper and lower bounds of blow-up rates have been obtained.
文摘Under both controllable and natural growth conditions, the spatial derivatives D_auof a solution u∈ L^2(0, T; H^1(Q , R^N)) ∩L~∞(0, T; L^2(Q , R^N)) (or∩L~∞(Q , R^N)) of a nonlinearparabolic system u_t^i - D_αA_i~α(x, t, u, Du) = B_i(x, t, u, Du), i = 1,…, N, (x, t) ∈Q,in fact, belong to L_(loc)~p(Q,R^N) for some p>2. Every solution of a quasilinear parabolic system u_t^i - D_α[A_(ij)^(αβ)(x, t, u)D_βu^j +α_i~α(x, t, u)] = B_i(x, t, u, Du), i = 1, …, Nis Holder continuous in an open set Q_1?Q with H^(n+2-p)(Q\Q_1)=0. If A_(ij)^(αβ)(x,t,u)=0 when j>i, and the growth of B_i(x, t, u, p) w. r. t. |p| is less than 2, then Q_1=Q. If A_(ij)^(αβ) and α_i~αare Holder continuous, then so are D_αu in Q_1.
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.
基金supported by the National Natural Science Foundation of China under Grant No.60607013.
文摘The two-dimensional（2D） compound parabolic concentrator＇s（CPC） characteristics are analyzed.It is shown that CPC＇s height is taller and its light collecting ability is stronger with the CPC＇s field of view decreasing when the bottom radius is unchanged.According to the ZEMAX analysis,CPC is good at collecting optical signal,and the antenna combining CPC with hemispherical lens can gather more optical signal than a single CPC or CPCs combined in series.The light propagation of scattering optical communication based on multiple scattering is simulated by Monte Carlo method,and the results show that using CPC as receiving antenna can strengthen communication system＇s signal collecting ability and increase its communication distance.
基金Natural Science Foundation of China (Grant Nos.11671369,11271340).
文摘This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation.Taking the finite dement pair Q11/Q01×Q10 as an example, a new mixed finite element method (FEM)is established and the r-independent superclose results of the original variable u in Hi-norm and the flux variable q=-a(u)■u in L^2- norm are deduced (τ is the temporal partition parameter).A key to our analysis is all error splitting technique,with which the time-discrete and the spatial-discrete systems are constructed,respectively.For the first system,tile boundedness of the temporal errors are obtained.For the second system,the spatial superclose results are presented unconditionally.while the previous literature always only obtain the convergent estimates or require certain time step conditions.Finally,some numerical results are provided to confirm the theoretical analysis,and show the efficiency of the proposed method.
文摘In 1988, Yu . A. Alkhutov and I. T. Mamedov discussed the solvability of the Dirichlet problem for linear uniformly parabolic equations with measurable coefficients where the coefficients satisfy the condition In this paper, we try to generalize the results of Alkhutov and Mamedov to nonlinear uni- formly parabolic systems of second order equations with measurable coefficients; moreover, we also discuss the solvability of the Neumann problem for the above systems.
基金supported by the National Natural Science Foundation of China (Grant No. 61573013)。
文摘This paper studies the problem of adaptive neural networks control(ANNC) for uncertain parabolic distributed parameter systems(DPSs) with nonlinear periodic time-varying parameter(NPTVP). Firstly, the uncertain nonlinear dynamic and unknown periodic TVP are represented by using neural networks(NNs) and Fourier series expansion(FSE), respectively. Secondly, based on the ANNC and reparameterization approaches, two control algorithms are designed to make the uncertain parabolic DPSs with NPTVP asymptotically stable. The sufficient conditions of the asymptotically stable for the resulting closed-loop systems are also derived. Finally, a simulation is carried out to verify the effectiveness of the two control algorithms designed in this work.
