Albert Retey has demonstrated in a similar situation that you can use "EventLocator" to detect an event in NDSolve. For example:

    NDSolve[{\!\(
    \*SubscriptBox[\(∂\), \(t\)]\(u[t, x]\)\) == 1/100 \!\(
    \*SubscriptBox[\(∂\), \(x, x\)]\(u[t, x]\)\) - u[t, x] \!\(
    \*SubscriptBox[\(∂\), \(x\)]\(u[t, x]\)\), 
        u[0, x] == Sin[2 π x], u[t, 0] == u[t, 1]}, u, {t, 0, 2}, {x, 
      0,
         1}, StepMonitor :> (laststep = thisstep; thisstep = t;
          stepsize = thisstep - laststep;), 
      Method -> {"EventLocator", "Event" :> (If[stepsize < 10^-4, 0, 1])}]

Then the integration will stop when the step size is less than 10^-4, and the variable `thisstep` will be the point you are looking for.