I use Linux all the time but I haven't a clue what the data in your pictures tells us. In any case I presume you asked a question here because the problem is in your Arduino program rather than in Linux.
Can you put some code in your program that prints messages at regular intervals from different parts of the program. Then when the program crashes you may get some idea where the cause of the problem lies. However if the problem is due to memory corruption due to the use of the String class I would expect the problem to occur in a somewhat random fashion.
...R