Convection-Diffusion Equations in Uniformly Local Lebesgue Spaces

In this paper, we establish the local existence and uniqueness of the mild solution to the Cauchy problem for convection-diffusion equation in n-dimensional Euclidean space with initial data in uniformly local function spaces \(L^r_{uloc,\rho}\)(\(\mathbb{R}^n\)). For the proof, we apply the uniformly local \(L^p_{uloc,\rho}\)(\(\mathbb{R}^n\)) - \(L^q_{uloc,\rho}\)(\(\mathbb{R}^n\)) estimate for the convolution operators got by Maekawa and Terasawa [1], and the Banach fixed point hypothesis.