TY - JOUR
T1 - A nonlinear semigroup approach to Hamilton-Jacobi equations–revisited
AU - Ni, Panrui
AU - Wang, Lin
N1 - Publisher Copyright:
© 2024
PY - 2024/9/15
Y1 - 2024/9/15
N2 - We consider the Hamilton-Jacobi equation H(x,Du)+λ(x)u=c,x∈M, where M is a connected, closed and smooth Riemannian manifold. The functions H(x,p) and λ(x) are continuous. H(x,p) is convex, coercive with respect to p, and λ(x) changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao [11] under the Tonelli conditions. In this paper, we consider more detailed structure of the viscosity solution set and large time behavior of the viscosity solution on the Cauchy problem. To the best of our knowledge, it is the first detailed description of the large time behavior of the HJ equations with non-monotone dependence on the unknown function.
AB - We consider the Hamilton-Jacobi equation H(x,Du)+λ(x)u=c,x∈M, where M is a connected, closed and smooth Riemannian manifold. The functions H(x,p) and λ(x) are continuous. H(x,p) is convex, coercive with respect to p, and λ(x) changes the signs. The first breakthrough to this model was achieved by Jin-Yan-Zhao [11] under the Tonelli conditions. In this paper, we consider more detailed structure of the viscosity solution set and large time behavior of the viscosity solution on the Cauchy problem. To the best of our knowledge, it is the first detailed description of the large time behavior of the HJ equations with non-monotone dependence on the unknown function.
KW - Aubry-Mather theory
KW - Contact Hamiltonian systems
KW - Hamiltonian systems
KW - Weak KAM theory
UR - http://www.scopus.com/inward/record.url?scp=85193985134&partnerID=8YFLogxK
U2 - 10.1016/j.jde.2024.05.039
DO - 10.1016/j.jde.2024.05.039
M3 - Article
AN - SCOPUS:85193985134
SN - 0022-0396
VL - 403
SP - 272
EP - 307
JO - Journal of Differential Equations
JF - Journal of Differential Equations
ER -