Space-time tradeoff lower bounds
Here are a few additional references. More can be found by looking at the papers that cite these.

Duris and Galil (1984) give a language in $P$ which requires $T^2 S \geq \Omega(n^3)$ on one-tape Turing machines with any constant number of read-write heads. Karchmer (1986) showed that the same lower bound holds for the element distinctness problem. 

Babai, Nisan, and Szegedy (1989) give a very natural language (generalized inner product) that is solvable in $O(n)$ time and $O(1)$ space on a $k+1$-head one-tape Turing machine, that requires $T S \geq \Omega(n^2)$ on any $k$-head one-tape Turing machine.

Ajtai (1999) shows time-space tradeoffs for deterministic random access machines computing element distinctness. In particular if $S \leq o(n)$, then $T \geq \omega(n)$. Subsequent work by Beame, Saks, Sun, and Vee (2000) proves time-space tradeoffs for randomized computations.

Santhanam (2001) showed that $TS \geq \Omega(n^2)$ holds for multitape Turing machines solving SAT, building on Cobham's analogous lower bound for PALINDROMES.