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Minimax Theory and its Applications 05 (2020), No. 2, 383--400
Copyright Heldermann Verlag 2020

On Almost Periodic Viscosity Solutions to Hamilton-Jacobi Equations

Evgeny Yu. Panov
Algebra and Analysis Department, Novgorod State University, Veliky Novgorod 173003, Russia;
and: Peoplesí Friendship University, Moscow 117198, Russia

We establish that a viscosity solution to a multidimensional Hamilton-Jacobi equation with Bohr almost periodic initial data remains to be spatially almost periodic and the additive subgroup generated by its spectrum does not increase in time. In the case of one space variable and a non-degenerate hamiltonian we prove the decay property of almost periodic viscosity solutions when time t goes to ∞. For convex hamiltonian we also provide another proof of this property using the Hopf-Lax-Oleinik formula. For periodic solutions the more general result is proved on unconditional asymptotic convergence of a viscosity solution to a traveling wave.

Keywords: Hamilton-Jacobi equations, viscosity solutions, almost periodic functions, long time behavior.

MSC: 35F21 35D40 35B15; 35B40.

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