Finite time blow up solutions for heat equations with Neumann boundary conditions on $\mathbb{R}_{+}^{4}$

We consider the nonlinear heat equations with Neumann boundary conditions $$\begin{cases} u_{t}=\Delta u & \text{in}\ \mathbb{R}_{+}^{4} \times(0, T) ,\\ -\frac{d u}{d x_{4}}(\tilde{x}, 0, t) \ =u^2(\tilde{x}, 0, t)& \text{in}\ \mathbb{R}^{3} \times(0, T). \end{cases}$$ We establish the existence of a finite-time blow-up solution. Specifically, for any sufficiently small $T>0$ and any $k$ distinct points $q_{1},\dots,q_{k}\in \mathbb{R}^{3}$, there exists an initial datum $u_{0}$ such that the corresponding solution $u(x,t)$ blows up exactly at $q_{1},\dots,q_{k}$ as $t\nearrow T$. Furthermore, when $t\nearrow T$, the solution admits the asymptotic profile $$u(x,t)=\sum_{j=1}^{k}U_{\mu_{j}(t),\xi_{j}(t)}(x)+Z_0^*(x)+o(1)\quad \text{as}~ t\nearrow T,$$ where $$U_{\mu_{j}(t),\xi_{j}(t)}(x):=\mu_{j}^{-1}(t) U\left(\frac{x-\xi_{j}(t)}{\mu_{j}(t)}\right),~ x\in \mathbb{R}_{+}^{4},$$ and $Z_{0}^{*}\in C_{0}^{\infty}(\mathbb{R}_{+}^{4})$ satisfying $$Z_{0}^{*}(q_{j},0)<0\quad \text{for all}\ j=1,\dots,k.$$ Here, $U(y)$ denotes the harmonic extension to $\mathbb{R}_{+}^{4}$ of the positive radially symmetric solution $\widetilde{U}$ to the fractional Yamabe problem $(-\Delta)^{\frac{1}{2}} \widetilde{U} = \widetilde{U}^{2}$ in $\mathbb{R}^{3}$. For some constants $\beta_{j}>0$, the scaling parameters $\mu{j}(t)$ and the translation parameters $\xi_{j}(t)$ satisfy $$\mu_{j}(t)=\beta_{j}\frac{|\log 2T|(T-t)}{|\log(T-t)|^{2}}(1 + o(1)) \to 0,~\xi_{j}(t)\to (q_{j},0)\quad \text{as} ~t\nearrow T.$$