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## 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.

### В книге

Providence: American Mathematical Society, 2005.