A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces
A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces
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DOI：
10.1016/j.anihpc.2015.06.005
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发表时间：
2014-07
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影响因子：
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通讯作者：
R. Laister;James C. Robinson;M. Sierżęga;Alejandro Vidal-L'opez
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文献类型：
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作者：
R. Laister;James C. Robinson;M. Sierżęga;Alejandro Vidal-L'opez
We consider the scalar semilinear heat equation u t− Δ u= f (u), where f:[0,∞)→[0,∞) is continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in L q (Ω) for all non-negative initial data u 0∈ L q (Ω), when Ω⊂ R d is a bounded domain with Dirichlet boundary conditions. For q∈(1,∞) this holds if and only if lim sup s→∞ s−(1+ 2 q/d) f (s)<∞; and for q= 1 if and only if∫ 1∞ s−(1+ 2/d) F (s) d s<∞, where F (s)= sup 1≤ t≤ s f (t)/t. This shows for the first time that the model nonlinearity f (u)= u 1+ 2 q/d is truly the ‘boundary case’when q∈(1,∞), but that this is not true for q= 1. The same characterisations hold for the equation posed on the whole space R d provided that lim sup s→ 0 f (s)/s<∞.
