Blow-up behavior for a nonlinear heat equation with a localized source in a ball |
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Authors: | Isamu Fukuda Ryuichi Suzuki |
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Affiliation: | Department of Mathematics, Faculty of Engineering, Kokushikan University, 4-28-1 Setagaya Setagaya-ku, Tokyo 154-8515, Japan |
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Abstract: | In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: ut=Δu+up+uq(x*,t) in B(R) where p,q>0,B(R)={x∈RN:|x|<R} and x*≠0. If max(p,q)>1, there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q.- (i)
- If or p=q>2, then single point blow-up occurs whenever solutions blow up.
- (ii)
- If 1<p<q, both phenomena, total blow-up and single point blow-up, occur depending on the initial data.
- (iii)
- If p?1<q, total blow-up occurs whenever solutions blow up.
- (iv)
- If max(p,q)?1, every solution exists globally in time.
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Keywords: | Single point blow-up Total blow-up Localized reaction Non-local problem |
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