TY - JOUR
T1 - Hamilton–Jacobi equations with their Hamiltonians depending Lipschitz continuously on the unknown
AU - Ishii, Hitoshi
AU - Wang, Kaizhi
AU - Wang, Lin
AU - Yan, Jun
N1 - Publisher Copyright:
© 2021 Taylor & Francis Group, LLC.
PY - 2022
Y1 - 2022
N2 - We study the Hamilton–Jacobi equations (Formula presented.) in M and (Formula presented.) in (Formula presented.) where the Hamiltonian (Formula presented.) 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.
AB - We study the Hamilton–Jacobi equations (Formula presented.) in M and (Formula presented.) in (Formula presented.) where the Hamiltonian (Formula presented.) 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.
KW - Cauchy problem
KW - Hamilton–Jacobi equation
KW - comparison principle
KW - semicontinuous solutions
UR - http://www.scopus.com/inward/record.url?scp=85117308358&partnerID=8YFLogxK
U2 - 10.1080/03605302.2021.1983598
DO - 10.1080/03605302.2021.1983598
M3 - Article
AN - SCOPUS:85117308358
SN - 0360-5302
VL - 47
SP - 417
EP - 452
JO - Communications in Partial Differential Equations
JF - Communications in Partial Differential Equations
IS - 2
ER -