Fujita Type Conditions to Heat Equation with Variable Source

• Published in: Acta Mathematicae Applicatae Sinica 2017-02, Vol.33 (1), p.63-68
• Main Authors: Liu, Yun-xia, Wang, Yun-hua, Zheng, Si-ning
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Continuity (mathematics); iii; Math Applications in Computer Science; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Theoretical; 整体解; 有界函数; 有限时刻爆破; 热方程; 相互作用; 非负解
• ISSN: 0168-9673; 1618-3932
• DOI: 10.1007/s10255-017-0635-8

This paper studies heat equation with variable exponent ut = △u ＋ Up（x） 4- Uq in RN × （0, T）, where p（x） is a nonnegative continuous, bounded function, 0 〈 p- = infp（x） ≤ p（x） ≤ supp（x） = p＋. It is easy to understand for the problem that all nontrivial nonnegative solutions must be global if and only if max{p＋,q} ≤1. Based on the interaction between the two sources with fixed and variable exponents in the model, some Fujita type conditions are determined that that all nontrivial nonnegative solutions blow up in finite time if 0 〈 q ≤ 1 with p＋ 〉 1, or 1 〈 q 〈 1 ＋2/N. In addition, if q 〉 1 ＋2/N, then （i） all solutions blow up in finite time with 0 〈 p- ≤ p＋ ≤ 1 ＋2/N; （ii） there are both global and nonglobal solutions for p- ≤ 1 ＋ 2/N; and （iii） there are functions p（x） such that all solutions blow up in finite time, and also functions p（x） such that the problem possesses global solutions when p＋〈 1＋2/N 〈 p＋.