Short time behavior near the boundary for the heat equation with a nonlinear boundary condition (Q1612565)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^N\) with a smooth boundary \(\partial\Omega\) and let \(n=n(x)\) denote the inner unit normal at \(x\in\partial\Omega\). Consider the heat equation \(u_t=\Delta u\) in \(\Omega\times(0,T)\) complemented by the nonlinear boundary condition \(\partial u/\partial n=\varphi(u)\) on \(\partial\Omega\times(0,T)\) and the initial condition \(u(x,0)\equiv M\). The authors study the asymptotic behavior of the solution \(u\) close to the boundary \(\partial\Omega\) for small time \(t\). More precisely, assuming \(x\in\partial\Omega\) and \(s,t>0\) they establish the \(\varepsilon\)-expansion of \(u(x+\varepsilon sn(x),\varepsilon^2t)\) up to the order two. The coefficients in this expansion depend only on \(M,\varphi,s,t\) and the principal curvatures of \(\partial\Omega\) at \(x\).