Hamilton–Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown

Abstract

We study the Hamilton–Jacobi equations $H(x,Du,u)=0$ in M and $\partial u/\partial t+H(x,D_{x}u,u)=0$ in $M\times (0,\infty ),$ where the Hamiltonian $H=H(x,p,u)$ depends Lipschitz continuously on the variable u. In the framework of the semicontinuous viscosity solutions due to Barron–Jensen, we establish the comparison principle, existence theorem, and representation formula as value functions for extended real-valued, lower semicontinuous solutions for the Cauchy problem. We also establish some results on the long-time behavior of solutions for the Cauchy problem and classification of solutions for the stationary problem.

Keywords:

• Cauchy problem
• comparison principle
• Hamilton–Jacobi equation
• semicontinuous solutions

2010 Mathematics Subject Classification:

• Primary 35F21
• Secondary 35D40
• 35B51
• 35B40
• 49H25

Additional information

Funding

H. Ishii was partially supported by the JSPS KAKENHI Grant Nos. JP16H03948, JP20K03688, JP20H01817, and JP21H00717; K. Wang was partially supported by NSFC Grant Nos. 11771283, 11931016; L. Wang was partially supported by NSFC Grant Nos. 11790273, 12122109; J. Yan was partially supported by NSFC Grant Nos. 11631006, 11790273.