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

TL;DR

This paper constructs solutions to a nonlinear heat equation with Neumann boundary conditions on  that blow up at specified points in finite time, providing detailed asymptotic profiles and scaling behaviors.

Contribution

It establishes the existence of finite-time blow-up solutions with prescribed blow-up points and detailed asymptotic profiles for a class of nonlinear heat equations with Neumann boundary conditions.

Findings

Solutions blow up exactly at specified points as time approaches T.

The asymptotic profile of solutions is characterized by a sum of scaled harmonic extensions plus a smooth correction.

Scaling parameters decay at a rate involving logarithmic factors as t approaches T.

Abstract

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