Problem that is only solvable in a given space
This kind of question is studied in the cryptography literature under the name "memory-bound computation" or "memory-hard function".  Such a function is one that can be computed efficiently using $K$ bytes of memory but takes much longer to compute if you have significantly less than $K$ bytes of memory.  So, I suggest that you take a look at cryptographic constructions for memory-hard functions.  It's an active area of research, and researchers focus more on practical complexity (concrete running time, not asymptotic running time) than on complexity theory, but you might find it useful.

See https://en.wikipedia.org/wiki/Memory_bound_function#Using_memory_bound_functions_to_prevent_spam for a high-level overview.