关于半线性热方程的防爆与防熄问题

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On the Problem of Preventing Blowing-up and Quenching for Semilinear Heat Equation

摘要: 本文研究初值问题
u_t=Δu+g(t)f(u)(t>0),u|_t=0=u₀(x)
和初边值问题
u_t=Δu+g(t,x)f(u)(t>0,x∈Ω),u|_t=0=u|_∂Ω=0
之解的整体存在性。如文献[6]中所作的那样,在非线性项中引进因子g(t)或g(t,x),是为了防止解的爆破或熄灭现象发生。本文的结果表明,文献[6]的两个定理中对f,g和u₀的大部分限制可以取消或者减弱;对g可以只要求它在f大时充分小;在一定条件下,控制初始状态即可避免爆破。

Abstract: In this paper, the global existence of solutions to the IVP
u_t=Δu+g(t)f(u)(t>0),u|_t=0=u₀(x)
and the IBVP
u_t=Δu+g(t,x)f(u)(t>0,x∈Ω),u|_t=0=u|_∂Ω=0
is investigated, As has been done in[6],the introduction of factors g(t) or g(t,x) in nonlinear term is to prevent the occurrence of blowing-up or quenching of solutions, It is shown in this paper that most of the restrictions on f, g and u₀ in the theorems of [6] may be cancelled or relaxed,that the smallness of g is required only fortlarge,and that under certain conditions controlling initial state can avoid blowing-up.

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