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 $$\mu_{ j}(t)=\beta_{ j}\frac{ |\log 2T|(T-t)}{ |\log(T-t)|^{ 2}}(1 + o(1)) \to 0,~\xi_{ j}(t)\to (q_{ j},0)\quad \text{ as} ~t\nearrow T.$$
Categories: math.AP