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 $$\{\begin{array}{ll}{u}_t=\Delta u& \text{in}{\mathbb{R}}_{+}^4\times (0,T),\\ -\frac{du}{d{x}_4}(\tilde{x},0,t)=u^2(\tilde{x},0,t)& \text{in}{\mathbb{R}}^3\times (0,T).\end{array}$$ 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 $$u(x,t)=\sum _{j=1}^k{U}_{{\mu }_j(t),{\xi }_j(t)}(x)+Z_0^{\ast }(x)+o(1)\text{as}t\nearrow T,$$ 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 $Z_0^{\ast }\in C_0^{\infty }({\mathbb{R}}_{+}^4)$ satisfying $$Z_0^{\ast }(q_j,0)<0\text{for all}j=1,\dots ,k.$$ Here, $U(y)$ denotes the harmonic extension to ${\mathbb{R}}_{+}^4$ of the positive radially symmetric solution $\tilde{U}$ to the fractional Yamabe problem $(-\Delta {)}^{\frac{1}{2}}\tilde{U}={\tilde{U}}^2$ in ${\mathbb{R}}^3$. For some constants ${\beta }_j>0$, the scaling parameters $\mu j(t)$ and the translation parameters ${\xi }_j(t)$ satisfy $${\mu }_j(t)={\beta }_j\frac{|\log 2T|(T-t)}{|\log (T-t){|}^2}(1+o(1))\to 0,{\xi }_j(t)\to (q_j,0)\text{as}t\nearrow T.$$

60 pages