Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces

We establish the local existence and the uniqueness of solutions of the heat equation with a nonlinear boundary condition for the initial data in uniformly local $L^r$ spaces. Furthermore, we study the sharp lower estimates of the blow-up time of the solutions with the initial data $\lambda \psi$ as $\lambda \to 0$ or $\lambda \to \mathrm{\infty }$ and the lower blow-up estimates of the solutions.