Existence of weak solutions to a convection–diffusion equation in a uniformly local lebesgue space

Abstract

We consider the local existence and the uniqueness of a weak solution of the initial boundary value problem to a convection–diffusion equation in a uniformly local function space \begin{document}$ L^r_{{\rm uloc}, \rho}( \Omega) $\end{document} , where the solution is not decaying at \begin{document}$ |x|\to \infty $\end{document} . We show that the local existence and the uniqueness of a solution for the initial data in uniformly local \begin{document}$ L^r $\end{document} spaces and identify the Fujita-Weissler critical exponent for the local well-posedness found by Escobedo-Zuazua [ 10 ] is also valid for the uniformly local function class.