This paper deals with the initial-boundary value mixed problems for nonlinear wave equations. By introducing the 'blowing-up facts K(u,u_i)', We may discuss the blowing up behaviours of solutions in finite tim...This paper deals with the initial-boundary value mixed problems for nonlinear wave equations. By introducing the 'blowing-up facts K(u,u_i)', We may discuss the blowing up behaviours of solutions in finite time to the mixed problems with respect to Neumann boundary and Dirichlet boundary for various nonlinear conditions and initial value conditions which usually meet.展开更多
In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t...In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t）=（u^p11v^p12）（1,t）,vx（1,t）=（u^p21v^p22）（1,t）,t∈（0,T）,u（x,0）=u0（x）,v（x,0）=v0（x）,x∈（0,1）We will prove that there exist two positive constants such that：c≤max x∈[0,1]u（x,t）（T-t）^r（l1-1）≤C,0〈t〈T,c≤max x∈[0,1] v（x,t）（T-t）^1/（t1-1）≤C,0〈t〈T.where l1=l21α/α2＋l22,r=α1/α2〉1,α1≤α2〈0.展开更多
It is generally known that the solutions of deterministic and stochastic differential equations （SDEs） usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventual...It is generally known that the solutions of deterministic and stochastic differential equations （SDEs） usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventually blow up or explode in finite time. If the drift and diffusion functions are globally Lipschitz, linear growth may still be experienced, as well as a possible blow-up of solutions in finite time. In this paper, a nonlinear scalar delay differential equation with a constant time lag is perturbed by a multiplicative Ito-type time - space white noise to form a stochastic Fokker-Planck delay differential equation. It is established that no explosion is possible in the presence of any intrinsically slow time - space white noise of Ito - type as manifested in the resulting stochastic Fokker- Planck delay differential equation. Time - space white noise has a role to play since the solution of the classical nonlinear equation without it still exhibits explosion.展开更多
The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory: u_(tt) - u_(xx) = 1/ Γ(1 - γ) ∫_0~t (t - s)^(-γ)|u(s)|~pds. The blow up result will be es...The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory: u_(tt) - u_(xx) = 1/ Γ(1 - γ) ∫_0~t (t - s)^(-γ)|u(s)|~pds. The blow up result will be established when p > 1 and 0 < γ < 1, no matter how small the initial data are, by introducing two test functions and a new functional.展开更多
Addressed here is the occurrence of point singularities which owe to the focusing of short or long waves, a phenomenon labeled dispersive blow-up. The context of this investigation is linear and nonlinear, strongly di...Addressed here is the occurrence of point singularities which owe to the focusing of short or long waves, a phenomenon labeled dispersive blow-up. The context of this investigation is linear and nonlinear, strongly dispersive equations or systems of equations. The present essay deals with linear and nonlinear Schrdinger equations, a class of fractional order Schrdinger equations and the linearized water wave equations, with and without surface tension. Commentary about how the results may bear upon the formation of rogue waves in fluid and optical environments is also included.展开更多
This paper is concerned with the finite time blow-up phenomena for the vector nonlinear Schrdinger equations with a magnetic field which describe the spontaneous generation of a magnetic field in a cold plasma in the ...This paper is concerned with the finite time blow-up phenomena for the vector nonlinear Schrdinger equations with a magnetic field which describe the spontaneous generation of a magnetic field in a cold plasma in the subsonic limit. After obtaining some a priori estimates,we prove under certain natural conditions that the solutions to the Cauchy problem of the vector nonlinear Schrdinger equations in two and three spatial dimensions blow up in a finite time. Assuming that a solution to the aforementioned vector nonlinear Schrdinger equations is radially symmetric with respect to spatial variables x,we show that if the initial energy is non-positive,then the solution blows up in three dimensions in a finite time.展开更多
A moving collocation method is proposed and implemented to solve time fractional differential equations. The method is derived by writing the fractional differential equation into a form of time difference equation. T...A moving collocation method is proposed and implemented to solve time fractional differential equations. The method is derived by writing the fractional differential equation into a form of time difference equation. The method is stable and has a third-order convergence in space and first-order convergence in time for either linear or nonlinear equations. In addition, the method is used to simulate the blowup in the nonlinear equations.展开更多
文摘This paper deals with the initial-boundary value mixed problems for nonlinear wave equations. By introducing the 'blowing-up facts K(u,u_i)', We may discuss the blowing up behaviours of solutions in finite time to the mixed problems with respect to Neumann boundary and Dirichlet boundary for various nonlinear conditions and initial value conditions which usually meet.
文摘In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered：{ut=uxx＋u^l 11v^l 12,vt=vxx＋u^l21v^l22,（x,t）∈（0,1）×（0,T）,ux（0,t）=0,vx（0,t）=0,t∈（0,T）,ux（1,t）=（u^p11v^p12）（1,t）,vx（1,t）=（u^p21v^p22）（1,t）,t∈（0,T）,u（x,0）=u0（x）,v（x,0）=v0（x）,x∈（0,1）We will prove that there exist two positive constants such that：c≤max x∈[0,1]u（x,t）（T-t）^r（l1-1）≤C,0〈t〈T,c≤max x∈[0,1] v（x,t）（T-t）^1/（t1-1）≤C,0〈t〈T.where l1=l21α/α2＋l22,r=α1/α2〉1,α1≤α2〈0.
文摘It is generally known that the solutions of deterministic and stochastic differential equations （SDEs） usually grow linearly at such a rate that they may become unbounded after a small lapse of time and may eventually blow up or explode in finite time. If the drift and diffusion functions are globally Lipschitz, linear growth may still be experienced, as well as a possible blow-up of solutions in finite time. In this paper, a nonlinear scalar delay differential equation with a constant time lag is perturbed by a multiplicative Ito-type time - space white noise to form a stochastic Fokker-Planck delay differential equation. It is established that no explosion is possible in the presence of any intrinsically slow time - space white noise of Ito - type as manifested in the resulting stochastic Fokker- Planck delay differential equation. Time - space white noise has a role to play since the solution of the classical nonlinear equation without it still exhibits explosion.
基金supported by the National Natural Sicence Foundation of China(Nos.11301489,11401367,11501273)the Natural Science Foundation of Zhejiang Province(Nos.LQ13A010013,LY14A010010)the Doctoral Fund of Ministry of Education of China(No.20133108120002)
文摘The present paper is devoted to studying the initial-boundary value problem of a 1-D wave equation with a nonlinear memory: u_(tt) - u_(xx) = 1/ Γ(1 - γ) ∫_0~t (t - s)^(-γ)|u(s)|~pds. The blow up result will be established when p > 1 and 0 < γ < 1, no matter how small the initial data are, by introducing two test functions and a new functional.
基金supported by the Agence Nationale de la Recherche, France (No. ANR-07-BLAN-0250)the University of Illinois at Chicago,the Wolfgang Pauli Institute in Vienna, the University of Illinois at Chicago and the Université de Paris 11
文摘Addressed here is the occurrence of point singularities which owe to the focusing of short or long waves, a phenomenon labeled dispersive blow-up. The context of this investigation is linear and nonlinear, strongly dispersive equations or systems of equations. The present essay deals with linear and nonlinear Schrdinger equations, a class of fractional order Schrdinger equations and the linearized water wave equations, with and without surface tension. Commentary about how the results may bear upon the formation of rogue waves in fluid and optical environments is also included.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171241, 10801102, 11071177)
文摘This paper is concerned with the finite time blow-up phenomena for the vector nonlinear Schrdinger equations with a magnetic field which describe the spontaneous generation of a magnetic field in a cold plasma in the subsonic limit. After obtaining some a priori estimates,we prove under certain natural conditions that the solutions to the Cauchy problem of the vector nonlinear Schrdinger equations in two and three spatial dimensions blow up in a finite time. Assuming that a solution to the aforementioned vector nonlinear Schrdinger equations is radially symmetric with respect to spatial variables x,we show that if the initial energy is non-positive,then the solution blows up in three dimensions in a finite time.
基金supported by National Natural Science Foundation of China (Grant No.10901027)
文摘A moving collocation method is proposed and implemented to solve time fractional differential equations. The method is derived by writing the fractional differential equation into a form of time difference equation. The method is stable and has a third-order convergence in space and first-order convergence in time for either linear or nonlinear equations. In addition, the method is used to simulate the blowup in the nonlinear equations.
