Hi Ravi,
My friend bought these speakers from Singapore as a gift for me and I am facing exactly the same issue. If I unplug the DC, the speaker turns off and yes there is a difference in sound volume when plugin with AUX cable if I use good quality AUX cable the difference is less.
On the box it says it comes with 1 Year International warranty, I have contacted the customer support and waiting for there directions on the same. They have asked for the bill, which I have shared.
Thanks,
Nitin