scholarly journals Life span of positive solutions for a semilinear heat equation with general non-decaying initial data

2011 ◽  
Vol 379 (2) ◽  
pp. 518-523 ◽  
Author(s):  
Tohru Ozawa ◽  
Yusuke Yamauchi
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Life Span ◽  
Positive Solutions ◽  
Semilinear Heat Equation

A semilinear heat equation with singular initial data

1998 ◽  
Vol 128 (4) ◽  
pp. 745-758 ◽  
Author(s):  
Minkyu Kwak
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Decay Rate ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Semilinear Heat Equation ◽  
Singular Initial Data ◽  
Self Similar ◽  
Image Position

We first prove existence and uniqueness of non-negative solutions of the equationin in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the formwhere g = ga satisfiesAfter uniqueness is proved, the asymptotic behaviour of solutions ofis studied. In particular, we show thatThe case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

Compactness and single-point blowup of positive solutions on bounded domains

1997 ◽  
Vol 127 (1) ◽  
pp. 47-65 ◽  
Author(s):  
Stathis Filippas ◽  
Frank Merle
Keyword(s):  
Initial Data ◽  
Positive Solutions ◽  
Single Point ◽  
Time Interval ◽  
Bounded Domains ◽  
General Criterion ◽  
Smooth Convex ◽  
Semilinear Heat Equation ◽  
Image Position ◽  
Boundary Solutions

This paper is concerned with the blowup of positive solutions of the semilinear heat equationwith zero boundary conditions. The domainΩis supposed to be smooth, convex and bounded. We first show that, under the assumption that the initial data are uniformly monotone near the boundary, solutions that exist on the time interval (0,Tform a compact family in a suitable topology. We then derive some localisation properties of these solutions. In particular, we discuss a general criterion, independent of the initial data, which in some cases ensures single-point blowup.

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