This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and suff...This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and sufficient conditions under which all solutions to have a finite time blow-up and the exact blow-up rates are established. It is proved that the blow-up will occur only at the boundary x = 1. The asymptotic behavior near the blow-up time is also studied.展开更多
We consider the semilinear heat equation with globally Lipschitz non-linearity involving gradient terms in a bounded domain of R^n. In this paper, we obtain explicit bounds of the cost of approximate controllability, ...We consider the semilinear heat equation with globally Lipschitz non-linearity involving gradient terms in a bounded domain of R^n. In this paper, we obtain explicit bounds of the cost of approximate controllability, i.e., of the minimal norm of a control needed to control the system approximately. The methods we used combine global Carleman estimates, the variational approach to approximate controllability and Schauder＇s fixed point theorem.展开更多
This paper deals with the blow up properties of solutions to semilinear heat equation u t- Δ u=u p in R N +×(0,T) with the nonlinear boundary condition -ο u ο x 1 = u q for x 1=0,t∈(0,T) ....This paper deals with the blow up properties of solutions to semilinear heat equation u t- Δ u=u p in R N +×(0,T) with the nonlinear boundary condition -ο u ο x 1 = u q for x 1=0,t∈(0,T) .It has been proved that if max( p,q) ≤1,every nonnegative solution is global.When min (p,q) >1 by letting α=1p-1 and β=12(q-1) it follows that if max (α,β)≥N2 ,all nontrivial nonnegative solutions are nonglobal,whereas if max (α,β)<N2 ,there exist both global and nonglobal solutions.Moreover,the exact blow up rates are established.展开更多
We prove the approxomate controllability and finite dimensional exact controllability of semilinear heat equation in R <SUP>N </SUP>with the same control by introducing the weighted Soblev spaces.
A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the fol...A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the following conditions for someα∈(0,1]:||∫_(0)^(t)e−(t−s)^(A)dW(s)||L^(2)(Ω;L^(2)(O))≤C^(t^(α/2))and||∫_(0)^(t)e−(t−s)^(A)dW(s)||_B^(∞)L^(2)(Ω:H^(α)(O))≤C..The conditions above are shown to be satisfied by both trace-class noises(withα=1)and one-dimensional space-time white noises(withα=1/2).The latter would fail to satisfy the conditions withα=1/2 if the stochastic Besov norm||·||B∞L^(2)(Ω;˙H^(α)(O))is replaced by the classical Sobolev norm||·||L^(2)(Ω;˙Hα(O)),and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation.In this paper,the convergence of a modified exponential Euler method,with a spectral method for spatial discretization,is proved to have orderαin both the time and space for possibly nonsmooth initial data in L^(4)(Ω;˙H^(β)(O))withβ>−1,by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t=0.展开更多
It is shown that any solution to the semilinear problem{u（x,0=）u0（x）〈1,x∈[-1,1] u（±1,t）=0,t∈（0,T）, ut=uxx＋δ（1-u）^-p（x,t）∈（-1,1） ×（0,T）either touches 1 in finite time or converges smooth...It is shown that any solution to the semilinear problem{u（x,0=）u0（x）〈1,x∈[-1,1] u（±1,t）=0,t∈（0,T）, ut=uxx＋δ（1-u）^-p（x,t）∈（-1,1） ×（0,T）either touches 1 in finite time or converges smoothly to a steady state as t -~ ~e. Some extensions of this result to higher dimensions are also discussed.展开更多
文摘This paper deals with the blow-up properties of solutions to semilinear heat equation ut-uxx= up in (0, 1) × (0, T) with the Neumann boundary condition ux(0, t) = 0, u:x1, t) = 1 on [0, T). The necessary and sufficient conditions under which all solutions to have a finite time blow-up and the exact blow-up rates are established. It is proved that the blow-up will occur only at the boundary x = 1. The asymptotic behavior near the blow-up time is also studied.
基金supported by the Natural Science Foundation of China (No.10371136,10771222)
文摘We consider the semilinear heat equation with globally Lipschitz non-linearity involving gradient terms in a bounded domain of R^n. In this paper, we obtain explicit bounds of the cost of approximate controllability, i.e., of the minimal norm of a control needed to control the system approximately. The methods we used combine global Carleman estimates, the variational approach to approximate controllability and Schauder＇s fixed point theorem.
文摘This paper deals with the blow up properties of solutions to semilinear heat equation u t- Δ u=u p in R N +×(0,T) with the nonlinear boundary condition -ο u ο x 1 = u q for x 1=0,t∈(0,T) .It has been proved that if max( p,q) ≤1,every nonnegative solution is global.When min (p,q) >1 by letting α=1p-1 and β=12(q-1) it follows that if max (α,β)≥N2 ,all nontrivial nonnegative solutions are nonglobal,whereas if max (α,β)<N2 ,there exist both global and nonglobal solutions.Moreover,the exact blow up rates are established.
基金the National Natural Science Foundation of ChinaGuangdong Province Natural Science Foundation of China (No.021765).
文摘We prove the approxomate controllability and finite dimensional exact controllability of semilinear heat equation in R <SUP>N </SUP>with the same control by introducing the weighted Soblev spaces.
基金supported by National Natural Science Foundation of China(Grant Nos.12071020,12131005 and U2230402)the Research Grants Council of Hong Kong(Grant No.Poly U15300519)an Internal Grant of The Hong Kong Polytechnic University(Grant No.P0038843,Work Programme:ZVX7)。
文摘A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the following conditions for someα∈(0,1]:||∫_(0)^(t)e−(t−s)^(A)dW(s)||L^(2)(Ω;L^(2)(O))≤C^(t^(α/2))and||∫_(0)^(t)e−(t−s)^(A)dW(s)||_B^(∞)L^(2)(Ω:H^(α)(O))≤C..The conditions above are shown to be satisfied by both trace-class noises(withα=1)and one-dimensional space-time white noises(withα=1/2).The latter would fail to satisfy the conditions withα=1/2 if the stochastic Besov norm||·||B∞L^(2)(Ω;˙H^(α)(O))is replaced by the classical Sobolev norm||·||L^(2)(Ω;˙Hα(O)),and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation.In this paper,the convergence of a modified exponential Euler method,with a spectral method for spatial discretization,is proved to have orderαin both the time and space for possibly nonsmooth initial data in L^(4)(Ω;˙H^(β)(O))withβ>−1,by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t=0.
基金Supported by National Natural Science Foundation of China (Grant No. 10801058)an Earmarked Grant for Research, Hong Kong and a self-determined Research Fund of CCNU from the Colleges’ Basic Research and Operation of MOE
文摘It is shown that any solution to the semilinear problem{u（x,0=）u0（x）〈1,x∈[-1,1] u（±1,t）=0,t∈（0,T）, ut=uxx＋δ（1-u）^-p（x,t）∈（-1,1） ×（0,T）either touches 1 in finite time or converges smoothly to a steady state as t -~ ~e. Some extensions of this result to higher dimensions are also discussed.
