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Regular version of the site

Book chapter

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

P. 161-179.
Sobolevski A., Khanin K., Khmelev D. V.

We construct an example of blow-up in a ”ow of min-plus linear operators arising as solution operators for a Hamilton…Jacobi equation @S/@t+|∇S| / + U(x, t) = 0, where > 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 de“ned 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/ . 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.

In book

Providence: American Mathematical Society, 2005.