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This paper studies the lower bound for the blow-up time $T^{*}$ of the heatequation $u_t=\Delta u$ in a bounded convex domain $\Omega$ in$\mathbb{R}^{N}(N\geq 2)$ with positive initial data $u_{0}$ and a localnonlinear Neumann boundary condition: the normal derivative $\partialu/\partial n=u^{q}$ on partial boundary $\Gamma_1\subseteq\partial\Omega$ forsome $q>1$, while $\partial u/\partial n=0$ on the other part. For any$\alpha<\frac{1}{N-1}$, we obtain a lower bound for $T^{*}$ which is of order$|\Gamma_{1}|^{-\alpha}$ as $|\Gamma_{1}|\rightarrow 0^{+}$, where$|\Gamma_{1}|$ represents the surface area of $\Gamma_{1}$. As$|\Gamma_{1}|\rightarrow 0^{+}$, this result significantly improves theprevious lower bound $\ln\big(|\Gamma_1|^{-1}\big)$ and is almost optimal indimension $N=2$, since the existing upper bound is of order $|\Gamma_{1}|^{-1}$as $|\Gamma_{1}|\rightarrow 0^{+}$. In addition, the optimal asymptotic orderof the lower bound for $T^{*}$ on $q$ (as $q\rightarrow 1^{+}$) and on $M_{0}$(as $M_{0}\rightarrow 0^{+}$) are obtained, where $M_{0}$ denotes the maximumof $u_{0}$.

Lower bounds for the blow-up time of the heat equation in convex domains with local nonlinear boundary conditions