Geodesic
Continuous generalized solution of the Hamilton--Jacobi equation with a noncoercive Hamiltonian
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M. T. Terekhin. Ryazan State University named for S. A. Yesenin, Ryazan, May 17-18, 2019. Part 2, Tome 186 (2020), pp. 144-151

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In this paper, we consider the Cauchy problem for the Hamilton–Jacobi equation with phase constraints arising in molecular biology. The problem has no classical solutions, and the Hamiltonian does not satisfy the conditions guaranteeing the existence of minimax or viscous generalized solutions. A new continuous generalized solution is obtained. We present sufficient conditions for the existence of a global solution preserving the structure given by the initial manifold. The behavior of such solutions is examined for large values of time.
Keywords: Hamilton–Jacobi equation, noncoercive Hamiltonian, generalized solution, phase constraint, method of characteristics. 
@article{INTO_2020_186_a18,
     author = {L. G. Shagalova},
     title = {Continuous generalized solution of the {Hamilton--Jacobi} equation with a noncoercive {Hamiltonian}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {144--151},
     publisher = {mathdoc},
     volume = {186},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_186_a18/}
}
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L. G. Shagalova. Continuous generalized solution of the Hamilton--Jacobi equation with a noncoercive Hamiltonian. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the All-Russian Scientific Conference «Differential Equations and Their Applications» dedicated to the 85th anniversary of Professor M. T. Terekhin. Ryazan State University named for S. A. Yesenin, Ryazan, May 17-18, 2019. Part 2, Tome 186 (2020), pp. 144-151. http://geodesic.mathdoc.fr/item/INTO_2020_186_a18/