Fujita type conditions to heat equation with variable source |
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Authors: | Yun-xia Liu Yun-hua Wang Si-ning Zheng |
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Institution: | 1.School of Mathematical Sciences,Dalian University of Technology,Dalian,China;2.School of Mathematical Sciences,Harbin Normal University,Harbin,China |
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Abstract: | This paper studies heat equation with variable exponent u t = Δu + up(x) + u q in ? 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 + \(\frac{2}{N}\). In addition, if q > 1 + \(\frac{2}{N}\), then (i) all solutions blow up in finite time with 0 < p? ≤ p+ ≤ 1 + \(\frac{2}{N}\); (ii) there are both global and nonglobal solutions for p? > 1 + \(\frac{2}{N}\); 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 + \(\frac{2}{N}\) < p+. |
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