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

By Matania Ben-Artzi, Philippe Souplet, Fred B. Weissler. 2002

127 citations recorded by the source corpus.

Abstract 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 u0, namely for u0 in Lebesgue spaces L q =L q ( R N ) , 1⩽q When p In the case a>0 and u0⩾0 (repulsive gradient term) we show that existence fails in all Lq spaces when p⩾2. When p Rather surprisingly, in the case a 2 under some additional assumption on u0), whereas nonexistence occurs for a large class of measure initial data if p>(N+2)/(N+1). In particular, a critical exponent for existence and uniqueness in the scale of Lq spaces appears if the gradient term is repulsive, while none occurs if it is absorbing.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/bbae6c3dc9e5539de211778528d85706ac58ee5a

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Citations in the corpus

Boundedness of global solutions for nonlinear parabolic equations involving gradient blow-up phenomena (2004, background): “A rather up-to-date list of references can be found in [6] (especially for the Cauchy problem, which has been more studied).”
Nonlinear equations with natural growth terms and measure data (2002, background): “Finally, let us mention that, in the linear case (p = 2), other existence and nonexistence results with gradient dependent lower order terms (absorbing or repulsive) and measure data are obtained in [1, 2, 6, 7] (see also the references cited therein).”
Single-Point Gradient Blow-up on the Boundary for Diffusive Hamilton-Jacobi Equations in Planar Domains (2010, background): “Existence, uniqueness and regularity of weak solutions of problem (1.5)‐(1.7) with irregular initial data are studied in [1,8, 11 ,22] and references therein.”
On the generalized Dirichlet problem for viscous Hamilton–Jacobi equations (2004, background): “Almost all these works deals with the equation in the whole space IRN and address questions on the existence, uniqueness and properties of either classical solutions (see for example Amour and Ben-Artzi[1], Ben-Artzi[10, 11], Gilding, Guedda and Kersner[19] and references therein) or solutions in th”
Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations (2004, background): “4 is proved in [7] while the second assertion is new.”