On variable Lebesgue spaces and generalized nonlinear heat equations

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 ${\mathcal{L}}_{\frac{nb}{2\alpha -⟨1{⟩}_{\gamma }}}^{p(\cdot )}({\mathbb{R}}^n,L^{\mathrm{\infty }}([0,T[))$. 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 $L^{p(\cdot )}([0,T],L^q({\mathbb{R}}^3))$.