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 $λψ$ as $λ\to 0$ or $λ\to\infty$ and the lower blow-up estimates of the solutions.

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