  In this work we address some questions concerning the Cauchy problem for a
generalized nonlinear heat equations considering as functional framework the
variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$. More precisely, by
mixing some structural properties of these spaces with decay estimates of the
fractional heat kernel, we were able to prove two well-posedness results for
these equations. In a first theorem, we prove the existence and uniqueness of
global-in-time mild solutions in the mixed-space $\mathcal{L}^{p(\cdot)}_{
\frac{nb}{2\alpha - \langle 1 \rangle_\gamma} } (\mathbb{R}^n,L^\infty([0,T[
))$. On the other hand, by introducing a new class of variable exponents, we
demonstrate the existence of an unique local-in-time mild solution in the space
$L^{p(\cdot)} \left( [0,T], L^{q} (\mathbb{R}^3) \right)$.
