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We construct an example of blow-up in a flow 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 defined on a time interval of length T, the absolute value of velocity for a Lagrangian minimizer can be as large as O (logT) 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.

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