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

Abstract: We consider the nonlinear heat equations with Neumann boundary conditions $$\begin{cases} u_{t}=Δu &amp; \text{in}\ \mathbb{R}<em>{+}^{4} \times(0, T) ,\ -\frac{d u}{d x</em>{4}}(\tilde{x}, 0, t) \ =u^2(\tilde{x}, 0, t)&amp; \text{in}\ \mathbb{R}^{3} \times(0, T). \end{cases}$$ We establish the existence of a finite-time blow-up solution. Specifically, for any sufficiently small $T&gt;0$ and any $k$ distinct points $q_{1},\dots,q_{k}\in \mathbb{R}<sup>{3}$, 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 $$u(x,t)=\sum_{j=1}^{{k}U_{μ<em>{j}(t),ξ</em>{j}(t)}(x)+Z_0^{*(x)+o(1)\quad}} \text{as}~ t\nearrow T,$$ where $$U_{μ<em>{j}(t),ξ</em>{j}(t)}(x):=μ<em>{j}^{-1}(t) U\left(\frac{x-ξ</em>{j}(t)}{μ<em>{j}(t)}\right),~ x\in \mathbb{R}</em>{+}<sup>{4},$$ and $$Z_{0}^{*}\in C_{0}<sup>{\infty}(\mathbb{R}_{+}<sup>{4})$$ satisfying $$Z_{0}^{{*}(q_{j},0)&lt;0\quad} \text{for all}\ j=1,\dots,k.$$ Here, $U(y)$ denotes the harmonic extension to $\mathbb{R}<em>{+}<sup>{4}$ of the positive radially symmetric solution $\widetilde{U}$ to the fractional Yamabe problem $$(-Δ)^{{\frac{1}{2}}} \widetilde{U} = \widetilde{U}<sup>{2}$$ in $\mathbb{R}<sup>{3}$. For some constants $β</em>{j}&gt;0$, the scaling parameters $μ{j}(t)$ and the translation parameters $ξ<em>{j}(t)$ satisfy $$μ</em>{j}(t)=β<em>{j}\frac{|\log 2T|(T-t)}{|\log(T-t)|^{2}}(1 + o(1)) \to 0,~ξ</em>{j}(t)\to (q_{j},0)\quad \text{as} ~t\nearrow T.$$