The local theory for viscous Hamilton–Jacobi equations in Lebesgue spaces

We consider viscous Hamilton–Jacobi equations of the form (VHJ) u t−Δu=a|∇u| p, x∈ R N, t>0, u(x,0)=u 0(x), x∈ R N, where a∈ R , a≠0 and p⩾1. We provide an extensive investigation of the local Cauchy problem for (VHJ) for irregular initial data u 0, namely for u 0 in Lebesgue spaces L q=L q( R N)...

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Bibliographic Details
Published in:Journal de mathématiques pures et appliquées 2002, Vol.81 (4), p.343-378
Main Authors: Ben-Artzi, Matania, Souplet, Philippe, Weissler, Fred B.
Format: Article
Language:English
Subjects:
Classical and quantum physics: mechanics and fields
Classical mechanics of discrete systems: general mathematical aspects
Critical exponents
Exact sciences and technology
Lebesgue spaces
Nonexistence
Nonlinear parabolic equations
Nonuniqueness
Physics
Viscous Hamilton–Jacobi equations
Well-posedness
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Summary:We consider viscous Hamilton–Jacobi equations of the form (VHJ) u t−Δu=a|∇u| p, x∈ R N, t>0, u(x,0)=u 0(x), x∈ R N, where a∈ R , a≠0 and p⩾1. We provide an extensive investigation of the local Cauchy problem for (VHJ) for irregular initial data u 0, namely for u 0 in Lebesgue spaces L q=L q( R N) , 1⩽ q
ISSN:0021-7824
DOI:10.1016/S0021-7824(01)01243-0