Well-posedness of the Cauchy problem for convection-diffusion equations in uniformly local Lebesgue spaces

Abstract

We consider the well-posedness of the Cauchy problem for convection-diffusion equations in uniformly local Lebesgue spaces $L^r_{\text{uloc}}(\mathbb R^n)$. In our setting, an initial function that is spatially periodic or converges to a nonzero constant at infinity is admitted. Our result is applicable to the one dimensional viscous Burgers equation. For the proof, we use the $L^p_{\text{uloc}}- L^q_{\text{uloc}}$ estimate for the heat semigroup obtained by Maekawa--Terasawa [20], the Banach fixed point theorem, and the comparison principle.