In this paper we deal with the initial boundary value problem for two classes of reaction diffusion systems with two source terms in bounded domain. Under some assumptions on the exponents and the initial data, applyi...In this paper we deal with the initial boundary value problem for two classes of reaction diffusion systems with two source terms in bounded domain. Under some assumptions on the exponents and the initial data, applying the comparison principle, the maximum prin- ciple and the supersolution-subsolution method, we prove the global existence and blow up of solutions. We also establish some upper blow up rates.展开更多
In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouvill...In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouville type theorems for semilinear parabolic problems.展开更多
We establish the existence, uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem -△pu=λ（x）u^θ-1-b（x）h（u）, in Ω,with boundary condition u = ＋∞ on δΩ, where ...We establish the existence, uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem -△pu=λ（x）u^θ-1-b（x）h（u）, in Ω,with boundary condition u = ＋∞ on δΩ, where Ω R^N （N≥2） is a smooth bounded domain, 1 〈 p 〈∞ λ（·） and b（·） are positive weight functions and h（u） ～ uq-1 as u → ∞. Our results extend the previous work [Z. Xie, J. Diff. Equ., 247 （2009）, 344-363] from case p = 2, λ is a constant and θ = 2 to case 1 〈 p 〈∞, A is a function and 1 （ 0 〈 θ 〈q 〉 p）; and also extends the previous work [Z. Xie, C. Zhao, J. Diff. Equ., 252 （2012）, 1776-1788], from case A is a constant and θ = p to case λ is a function and 1 〈 θ 〈 q （ 〉 p）. Moreover, we remove the assumption of radial symmetry of the problem and we do not require h（·） is increasing.展开更多
In this paper, we investigate the blow-up behavior of solutions of a parabolic equation with localized reactions. We completely classify blow-up solutions into the total blow-up case and the single point blow-up case,...In this paper, we investigate the blow-up behavior of solutions of a parabolic equation with localized reactions. We completely classify blow-up solutions into the total blow-up case and the single point blow-up case, and give the blow-up rates of solutions near the blow-up time which improve or extend previous results of several authors. Our proofs rely on the maximum principle, a variant of the eigenfunction method and an initial data construction method.展开更多
基金supported by the National Natural Science Foundation of China(11471087)the China Postdoctoral Science Foundation(2013M540270)+1 种基金the Heilongjiang Postdoctoral Foundation(LBH-Z13056,LBHZ15036)the Fundamental Research Funds for the Central Universities
文摘In this paper we deal with the initial boundary value problem for two classes of reaction diffusion systems with two source terms in bounded domain. Under some assumptions on the exponents and the initial data, applying the comparison principle, the maximum prin- ciple and the supersolution-subsolution method, we prove the global existence and blow up of solutions. We also establish some upper blow up rates.
文摘In this paper, we derive an upper bound estimate of the blow-up rate for positive solutions of indefinite parabolic equations from Liouville type theorems. We also use moving plane method to prove the related Liouville type theorems for semilinear parabolic problems.
基金Acknowledgments Research is partly supported by the National Science Foundation of China （10701066 ＆ 10971087）.
文摘We establish the existence, uniqueness and the blow-up rate of the large positive solution of the quasi-linear elliptic problem -△pu=λ（x）u^θ-1-b（x）h（u）, in Ω,with boundary condition u = ＋∞ on δΩ, where Ω R^N （N≥2） is a smooth bounded domain, 1 〈 p 〈∞ λ（·） and b（·） are positive weight functions and h（u） ～ uq-1 as u → ∞. Our results extend the previous work [Z. Xie, J. Diff. Equ., 247 （2009）, 344-363] from case p = 2, λ is a constant and θ = 2 to case 1 〈 p 〈∞, A is a function and 1 （ 0 〈 θ 〈q 〉 p）; and also extends the previous work [Z. Xie, C. Zhao, J. Diff. Equ., 252 （2012）, 1776-1788], from case A is a constant and θ = p to case λ is a function and 1 〈 θ 〈 q （ 〉 p）. Moreover, we remove the assumption of radial symmetry of the problem and we do not require h（·） is increasing.
基金Supported by the National Natural Science Foundation of China(No.10601011 and 10701024)
文摘In this paper, we investigate the blow-up behavior of solutions of a parabolic equation with localized reactions. We completely classify blow-up solutions into the total blow-up case and the single point blow-up case, and give the blow-up rates of solutions near the blow-up time which improve or extend previous results of several authors. Our proofs rely on the maximum principle, a variant of the eigenfunction method and an initial data construction method.
