Getting married for a second time

I am a divorced muslim woman with 2 children (boy & girl). The children live with me & they spend every alternate weekend with their father.
Their father contributes towards the financial maintenance of both the children.
The house we live in belongs to me & is fully paid for.

After careful consideration I intend to get married again.
The person is married with 2 children (boy & girl).

I willingly accept the following :
His ‘first’ home remains his main home / domicilium
. He will provide me with basic living expenses – I am aware that he will spend a lot more money on his ‘first’ family.
I will continue to live in my own home.
I know & accept that he will spend a lot more time with his ‘first’ family.
I dont make any demand on his finance and time.

I am aware I am entitled to equal rights but I willingly relinquish as above.
I consider it necessary for both of us.

If this is ok please also advise what happens to our children & material assets on the death of either of us?