The Blow-Up Rate for a Non-Scaling Invariant Semilinear Heat Equation

We consider the semilinear heat equation ∂ t u - Δ u = f ( u ) , ( x , t ) ∈ R N × [ 0 , T ) , ( 1 ) with f ( u ) = | u | p - 1 u log a ( 2 + u 2 ) , where p > 1 is Sobolev subcritical and a ∈ R . We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the loga...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2022-04, Vol.244 (1), p.87-125
Main Authors: Hamza, Mohamed Ali, Zaag, Hatem
Format: Article
Language:English
Subjects:
Analysis of PDEs
Classical Mechanics
Complex Systems
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Fluid- and Aerodynamics
Mathematical and Computational Physics
Mathematics
Physics
Physics and Astronomy
Theoretical
Thermodynamics
Upper bounds
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Summary:We consider the semilinear heat equation ∂ t u - Δ u = f ( u ) , ( x , t ) ∈ R N × [ 0 , T ) , ( 1 ) with f ( u ) = | u | p - 1 u log a ( 2 + u 2 ) , where p > 1 is Sobolev subcritical and a ∈ R . We first show an upper bound for any blow-up solution of (1). Then, using this estimate and the logarithmic property, we prove that the exact blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), namely u ′ = | u | p - 1 u log a ( 2 + u 2 ) . In other words, all blow-up solutions in the Sobolev subcritical range are Type I solutions. To the best of our knowledge, this is the first determination of the blow-up rate for a semilinear heat equation where the main nonlinear term is not homogeneous.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-022-01760-w