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Fujita type conditions to heat equation with variable source

Fujita type conditions to heat equation with variable source This paper studies heat equation with variable exponent u t = Δu + u p(x) + u q in ℝ N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p − = inf p(x) ≤ p(x) ≤ sup p(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 + $$\frac{2}{N}$$ 2 N . In addition, if q > 1 + $$\frac{2}{N}$$ 2 N , then (i) all solutions blow up in finite time with 0 < p − ≤ p + ≤ 1 + $$\frac{2}{N}$$ 2 N ; (ii) there are both global and nonglobal solutions for p − > 1 + $$\frac{2}{N}$$ 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 + $$\frac{2}{N}$$ 2 N < p +. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

Fujita type conditions to heat equation with variable source

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-017-0635-8
Publisher site
See Article on Publisher Site

Abstract

This paper studies heat equation with variable exponent u t = Δu + u p(x) + u q in ℝ N × (0, T), where p(x) is a nonnegative continuous, bounded function, 0 < p − = inf p(x) ≤ p(x) ≤ sup p(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 + $$\frac{2}{N}$$ 2 N . In addition, if q > 1 + $$\frac{2}{N}$$ 2 N , then (i) all solutions blow up in finite time with 0 < p − ≤ p + ≤ 1 + $$\frac{2}{N}$$ 2 N ; (ii) there are both global and nonglobal solutions for p − > 1 + $$\frac{2}{N}$$ 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 + $$\frac{2}{N}$$ 2 N < p +.

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Mar 15, 2017

References