Learn Arabic (Modern Standard): ثمة مشكلة كبيرة في هذا التقرير الأكاديمي. - There is a big problem in this academic report.

ثمة means 'there is' or 'there are', just like هناك. The difference is that ثمة is much more formal and mostly found in written Modern Standard Arabic. Grammatically, it acts as a fronted predicate, meaning the noun that follows it (مشكلة) is the actual subject of the sentence.

Why is the word مشكلة considered the subject here if it comes after ثمة?

In Arabic equational sentences, an indefinite subject usually cannot start the sentence. To say 'there is a problem', the formal locative adverb ثمة is brought to the front as the predicate, pushing the indefinite subject مشكلة to a delayed position. Because it is the subject, it takes the nominative case (مشكلةٌ).

Could I drop the definite article and write في هذا تقرير أكاديمي instead?

No, that would change the meaning completely. When you want a demonstrative phrase like 'this academic report', the noun following هذا must take the definite article الـ, as in هذا التقرير. Without the الـ, the phrase هذا تقرير means 'this is a report', which breaks the grammar of the sentence.

What case ending does the adjective الأكاديمي take at the end of this sentence?