Do in future and in the future imply different meanings? If so, using which one is grammatically correct?
He would be more careful in future.
They plan on getting married in the near future.
In the first sentence, "in future" means "from now on"; the NOAD says its use is chiefly British.
In the second sentence, "in the future" means "the time or a period of time following the moment of speaking or writing."
In future is commonly used in British English and is perfectly correct but has a different meaning than in the future.
In the future refers to an unspecified point in time, while in future means from now on.
It'a shame I missed you when I popped round to see you yesterday. I'll ring up beforehand in future.
In the future people will look back at the mobile phones we use today and laugh at their simplicity.
Both forms are possible. "In future" is often used in reprimands:
In future please sign and date your letters
It is very general, talking about all times from now onwards. You can find many examples on Google (search for "in future please take care").
"In the future" is more neutral and more specific:
In the future I would like to go to Australia, but now I don't have enough money.
The difference is only in whether future is a noun or an adjective.
In the future, please be more careful and Please be more careful in the future both use future as a noun.
In future activities, please be more careful uses future to modify the noun activities.
To my knowledge is it not correct to say in future in the place of in the future.
As far as grammar goes the difference between in future and in the future is contextual. One usually comes across phrases like in future endeavours or in future meets where future serves as an adjective. But if the reference is only to the times ahead,in the future would be correct and not in future. Thus,it would be correct to say: Be more careful in the future. or There will come a time in the future when there will be no petroleum left on the planet. and incorrect to go with the other. However,the spoken word in Britain and many former colonies where the influence of British English is enduring,it's not uncommon to come across simple notes that use in future. Perhaps they find it more easy on their ears.Nonetheless,grammatically wanting.