This question is complementary to question 1 asked by @Jesse here. It is about the case when we don't find a significant result when performing the log-rank test and when the proportional hazard assumption does not hold. In his/her answer to Jesse, @Aniko has started to answer my present question as follows: If it does not reject, then you have to worry about the proportionality of hazards and power. I totally understand we should worry, but I'd like to know how we should handle our anxiety then.
The log-rank test is always "valid" even if the proportional hazard assumption does not hold, in the following sense: under the null hypothesis $H_0$ (equality of hazard rates), the probability to reject achieves (approximately or asymptotically) the nominal significance level, that is, the power is low under $H_0$. But when the proportional hazard assumption holds true, then the behavior of the power under $H_1$ is clear because it should be an increasing function of the proportionality constant $k$.
But when this proportional hazard assumption does not hold, are there some typical "$H_1$ situations" for which it is known that the power is still low ? And for a given dataset, are there some visual methods to check whether we likely are in such a situation ?