﻿ 一类半线性抛物方程的解的爆破性质研究 Research on the Blasting Solution of a Class of Semilinear Parabolic Equations

Vol.06 No.01(2017), Article ID:19531,10 pages
10.12677/AAM.2017.61002

Research on the Blasting Solution of a Class of Semilinear Parabolic Equations

Liang Zhang1, Jianjun Li2

1College of Science, Inner Mongolia University of Technology, Hohhot Inner Mongolia

2College of Science, Liaoning Technical University, Fuxin Liaoning

Received: Dec. 20th, 2016; accepted: Jan. 9th, 2017; published: Jan. 16th, 2017

ABSTRACT

This article uses the variable index function to research the local solution of a class of semilinear parabolic equations of the blasting quality. In the limited domain of homogeneous Dirichlet condition, variable index function is the non-negative and bounded. With boundary condition of exponential function, we can define the local solution of semilinear parabolic equation at the condition of big initial data and arbitrary initial data of blasting conditions. Finally, we summarize all blasting properties about equations, and it was proved that the equation satisfies a Fujita type of conclusion; namely equation solution can be blasting in the variable index function and the size of domain that conditions of the nontrivial initial data are determinated.

Keywords:Semilinear Parabolic Equations, Variable Index Function, The Conclusion of the Fujita

1内蒙古工业大学理学院，内蒙古 呼和浩特

2辽宁工程技术大学理学院，辽宁 阜新

1. 引言

2. 方程介绍

(1.1)

(1.2)

3. 方程与解的处理

(2.1)

(2.2)

(2.3)

(2.1)，(2.2)，(2.3)一直遵循：

(2.2)中的满足利普希兹条件，(2.3)的非负解具有存在性和唯一性 [6] 。

(2.4)

，定义

，我们得到，因此：

，则

(2.5)

(2.6)

(2.7)

4. 大初始数据的爆破

(i)对于

(ii)对于

(3.1)

(3.2)

(3.3)

(3.4)

，(3.4)式的唯一解表示为：

(3.5)

(3.5)等式两侧同乘，然后两侧同时从到1进行积分。我们可以得到：

(3.6)

5. 任意初始数据的爆破

(4.1)

(4.2)

(4.3)

(4.4)

Research on the Blasting Solution of a Class of Semilinear Parabolic Equations[J]. 应用数学进展, 2017, 06(01): 10-19. http://dx.doi.org/10.12677/AAM.2017.61002

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