A blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain

Khanin, Konstantin; Khmelev, Dmitry; Sobolevskii, Andrei

A blow-up phenomenon in the Hamilton-Jacobi equation in an unbounded domain

Authors

Khanin, Konstantin

Khmelev, Dmitry

Sobolevskii, Andrei

Description

We construct an example of blow-up in a flow of min-plus linear operators arising as solution operators for a Hamilton-Jacobi equation with a Hamiltonian of the form |p|^alpha+U(x,t), where alpha>1 and the potential U(x,t) is uniformly bounded together with its gradient. The construction is based on the fact that, for a suitable potential defined on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O((log T)^(2-2/alpha)). We also show that this growth estimate cannot be surpassed. Implications of this example for existence of global generalized solutions to randomly forced Hamilton-Jacobi or Burgers equations are discussed.
19 pages, no figures; based on a talk given at the workshop "Idempotent Mathematics and Mathematical Physics" at the E. Schroedinger Institute for Mathematical Physics in Vienna in February 2003. A dedication indicating the untimely death of Dmitry Khmelev is added