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Yuxiang Li Weibing Deng Chunhong Xie 《Proceedings of the American Mathematical Society》2002,130(12):3661-3670
The initial-boundary value problems are considered for the strongly coupled degenerate parabolic system
in the cylinder , where is bounded and are positive constants. We are concerned with the global existence and nonexistence of the positive solutions. Denote by the first Dirichlet eigenvalue for the Laplacian on . We prove that there exists a global solution iff .
in the cylinder , where is bounded and are positive constants. We are concerned with the global existence and nonexistence of the positive solutions. Denote by the first Dirichlet eigenvalue for the Laplacian on . We prove that there exists a global solution iff .
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We consider the Cauchy problem for a doubly nonlinear degenerate parabolic equation with nonlocal source under the assumption
that the initial function is integrable. We establish the existence and nonexistence of time-global solutions of the problem.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1443–1464, November, 2005. 相似文献
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ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionInthispaper,weconsiderthefollowinginitial--boundaryvalueproblemwhereQ~fix(o,co),aQ=aflx(o,co),fiisaboundeddomaininEuclideanspaceR"(n22)withsmoothboundaryonandac=(u.,,'Iu..)denotesthegradientoffunctionu(x).Weassumethefunctionsal(x,t,u,p)(i=1,2,',n)anda(x,t,u,p)arelocallyH5ldercontinuousonfix(0,co)suchthatwherealtuandparepositiveconstants,m,aZIa3.hi,b2,alIadZ20,or321areconstants,m*E[0,m 2),hi16z/0,afl m*/… 相似文献
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Shaohua Chen 《Journal of Differential Equations》2008,245(4):1112-1136
The author discusses the degenerate and quasilinear parabolic system
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Impulsive effects on global existence of solutions for degenerate semilinear parabolic equations 总被引:4,自引:0,他引:4
Let q be a nonnegative real number, and λ and σ be positive constants. This article studies the following impulsive problem: for n = 1, 2, 3,…, . The number λ* is called the critical value if the problem has a unique global solution u for λ < λ*, and the solution blows up in a finite time for λ > λ*. For σ < 1, existence of a unique λ* is established, and a criterion for the solution to decay to zero is studied. For σ > 1, existence of a unique λ* and three criteria for the blow-up of the solution in a finite time are given respectively. It is also shown that there exists a unique T* such that u exists globally for T> T*, and u blows up in a finite time for T < T*. 相似文献
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This paper deals with the global existence and nonexistence of solutions of the second-order nonlinear differential equation (φ(x′))′+λφ(x)=0 satisfying x(0)=x0 and x′(0)=x1, where λ is a positive parameter and φ:(−ρ,ρ)→(−σ,σ) with 0<ρ?∞ and 0<σ?∞ is strictly increasing odd bijective and continuous on (−ρ,ρ). Necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic. Examples are given to illustrate our main result. Finally, a nonexistence result for the equation with a damping term is discussed as an application to our result. 相似文献
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Weibing Deng Yuxiang Li Chunhong Xie 《Proceedings of the American Mathematical Society》2003,131(5):1573-1582
This paper establishes a new criterion for global existence and nonexistence of positive solutions of the non-local degenerate parabolic system
0, \end{align*}">
with homogeneous Dirichlet boundary conditions, where is a bounded domain with a smooth boundary and are positive constants. For all initial data, it is proved that there exists a global positive solution iff , where is the unique positive solution of the linear elliptic problem
0, \end{align*}">
with homogeneous Dirichlet boundary conditions, where is a bounded domain with a smooth boundary and are positive constants. For all initial data, it is proved that there exists a global positive solution iff , where is the unique positive solution of the linear elliptic problem
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We establish local existence and comparison for a model problem which incorporates the effects of non-linear diffusion, convection and reaction. The reaction term to be considered contains a non-local dependence, and we show that local solutions can be obtained via monotone limits of solutions to appropriately regularized problems. Utilizing this construction, it is further shown that, under conditions of either ‘weak reaction’ or ‘sufficiently small’ initial mass, solutions exist for all time. Finally, we provide an alternative analysis of global existence and investigate blow up in finite time for the case of power law diffusion and convection. These results show the extent to which the assumption of weak reaction may be relaxed and still obtain global existence. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd. 相似文献
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This article deals with the degenerate parabolic system with nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve and the critical Fujita curve for the problem. Especially for the blow-up case, it is rather technical. It comes from the construction of the so-called Zel’dovich-Kompaneetz-Barenblatt profile. 相似文献
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Jing Li Jingxue Yin Chunhua Jin 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,20(1):835-847
This paper is concerned with the existence of nonnegative continuous solutions for the Cauchy problem of a class of fully
nonlinear degenerate parabolic equations. 相似文献
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Jing Li Jingxue Yin Chunhua Jin 《Zeitschrift für Angewandte Mathematik und Physik (ZAMP)》2010,61(5):835-847
This paper is concerned with the existence of nonnegative continuous solutions for the Cauchy problem of a class of fully nonlinear degenerate parabolic equations. 相似文献