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 a unique local-in-time mild solution in the space \({L}^{p(\cdot )} ([0,{T}], {L}^{q} (\mathbb {R}^n) )\) .
