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

Yan Zi-qian

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 1986 > 7(8): 713-718

Yan Zi-qian. On the Problem of Preventing Blowing-up and Quenching for Semilinear Heat Equation[J]. Applied Mathematics and Mechanics, 1986, 7(8): 713-718.

Citation:

Yan Zi-qian. On the Problem of Preventing Blowing-up and Quenching for Semilinear Heat Equation[J]. Applied Mathematics and Mechanics, 1986, 7(8): 713-718.

Citation:

Yan Zi-qian. On the Problem of Preventing Blowing-up and Quenching for Semilinear Heat Equation[J]. Applied Mathematics and Mechanics, 1986, 7(8): 713-718.

PDF( 365 KB)

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

Yan Zi-qian

Department of Mathematics, Jilin University, Jilin

Received Date: 1983-05-03
Publish Date: 1986-08-15

Abstract

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.

FullText(HTML)

References(8)

References

[1]	Kaplan, S,On the growth of solutions of quasi-linear parabolic equations, Comm,Pure Appl, Math,1g (1983),305-330.
[2]	Fujita, H,On the blowing up of solutions of the Cauchp problem for u_t=Δu+u^1+u,J.Fac, Sci, Univ, Tokpo, Sect, I,13(1966),109-124.
[3]	Hayakawa, K,On nonexistence of global solutions of some semilinear parabolic differential equations, Proc, Jopan Acad,49(1973),503-505.
[4]	Kawarada, H,On solutions of initial-boundary problem for u_t=u_xx+1/(1-u),Publ,RIMS,Kyoto Univ,10 (1975),729-736.
[5]	Acker, A, and W, Walter, The quenching problem for nonlinear parabolic differential equations, Lect, Notes in Math,Springer-Vcrlag,1-2(1976),564.
[6]	陈庆益,关于半线性热方程的爆破及熄灭问题,数学物理学报,2 (1982),17-23.
[7]	Fife,P, C,Mathematical aspects of reacting and diffusing systems, Lect, Notes in Biomathematics,Springer-Verlag(1979),28.
[8]	Weissler,F. B,Existence and-nonexistence of global solutions for a semilinear heat equation,Isreal J.Math,38 (1981),29-40.