两个都可以。
定冠词the，是指你说话语境时间的上下周。
例句：
in the next week
'We have a significant chance to dramatically reduce the oil that's being released into theenvironment and maybe shut the well in altogether in the next week,' he said.
我们有一个可以显著减少石油泄漏出来的重大机会，也许我们会在下一周关闭掉所有的油井。
on the next week
Our most successful students are those who have managed to consistently learn in shorterperiods of time over months, versus binge learning one week, and skipping out on the next week.
最成功的学生是那些能够在较短的时间内持续学习几个月，比那些学习一周，休息一周的学员相比，效果更好。
扩展资料：
in与on的用法区别
1、表示时间，注意以下用法：
泛指一般意义的上午、下午或晚上以及月或年等较长的时间，一般用 in：
We watch TV in the evening. 我们晚上看电视。
He went to Japan in 1946. 他于 1946 去了日本。
若表示星期几或某一特定的日期，则用 on：
He left here on the fifth of May. 他于 5 月 5 日离开这儿。
We have English class on Monday. 我们在周一有英语课
参考资料来源：