N. N. Subbotina, L. G. Shagalova, “The construction of a continuous generalized solution for the Hamilton–Jacobi equations with state constraints”, Izv. IMI UdGU, 2015, Issue 2(46),Pages <nobr>193

The construction of a continuous generalized solution for the Hamilton–Jacobi equations with state constraints

N. N. Subbotina^ab, L. G. Shagalova^a

^a Institute of Mathematics and Mechanics named after N.N. Krasovskii, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620219, Russia
^b Institute of Mathematics and Computer Science, Ural Federal University named after the first President of Russia B. N. Yeltsin, pr. Lenina, 51, Yekaterinburg, 620083, Russia

Abstract: We consider a boundary value problem with state constraints for a nonlinear non-coercive Hamilton–Jacobi equation. We introduce a new definition of continuous generalized solution of the problem and apply this definition to nonlinear non-coercive equation arising in molecular biology. The construction for generalized solution with additional requirements to structure is provided for this equation. Connections with viscosity generalized solutions are discussed. Results of computer simulations are exposed.

Keywords: Hamilton–Jacobi equation, generalized solution, viscosity solution, minimax solution, state constraints, method of characteristics.

UDC: 517.952, 517.97

MSC: 35D40, 35F21, 9L25

Received: 05.10.2015