It is known that $\mathrm{DTIME}[n]\subseteq \mathrm{DSPACE}[n/\log n]$. Therefore, there are languages in $\mathrm{DSPACE}[n]$ which are not in $\mathrm{DTIME}[o(n\log n)]$.Are there examples of natural problems which are in $\mathrm{DSPACE}[n]$ but not in $\mathrm{DTIME}[o(n\log n)]$? Are there examples of natural problems that are in $\mathrm{NSPACE}[n]$ but not in $\mathrm{DTIME}[o(n\log n)]$? By natural I mean problems which are not specifically designed to meet the specifications of the question. Clearly, using diagonalization arguments one can cook up problems that are in $\mathrm{DSPACE}[n]$ but not in $\mathrm{DTIME}[o(n\log n)]$. Note that the classes DTIME, DSPACE and NSPACE above are defined with respect to multitape turing machines. Obs 1:  I'm mostly interested in lower bounds of the form $\Omega(n\log n)$ or even $\Omega(n\log \log n)$ for Multitape turing machines. But weaker lower bounds would also be welcome. For instance, it has been shown in this paper, that satisfiability of succint QBF-formulas with a constant number of alternations require time $\Omega(n\log^* n)$, where $\log^* n$ is the iterated logarithm function. Although $\log^* n$ grows extremely slowly, this lower bound has the advantage that it works not only for multitape TMs but also for more general models, such as $d$-dimensional TM's, random access TM', and alike.
Natural Problems NSPACE[n] but not in DTIME[n]
lower bounds;complexity