Convection-Diffusion Equations in Uniformly Local Lebesgue Spaces
Mahfuza Khatun
Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh.
Md. Zahidul Islam
Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh.
Md. Rabiul Haque *
Department of Mathematics, University of Rajshahi, Rajshahi 6205, Bangladesh.
*Author to whom correspondence should be addressed.
Abstract
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.
Keywords: Convection-diffusion equation, uniformly local Lebesgue spaces, mild solutions, scaling invariance, well-posedness