  Our main purpose is to use a new condition, $\alpha$-local nondeterminism,
which is an alternative to the classical local nondeterminism usually utilized
in the Gaussian framework, in order to investigate Besov regularity, in the
time variable $t$ uniformly in the space variable $x$, for local times $L(x,
t)$ of a class of continuous processes. We also extend the classical Adler's
theorem [1, Theorem 8.7.1] to the Besov spaces case. These results are then
exploited to study the Besov irregularity of the sample paths of the underlying
processes. Based on similar known results in the case of the bifractional
Brownian motion, we believe that our results are sharp. As applications, we get
sharp Besov regularity results for some classical Gaussian processes and the
solutions of systems of non-linear stochastic heat equations. The Besov
regularity of their corresponding local times is also obtained.
