Finite time blow up solutions for heat equations with Neumann boundary conditions on $\mathbb{R}_{+}^{4}$
Xiang Fang, Juncheng Wei, Youquan Zheng
TL;DR
This work studies the nonlinear parabolic problem with a fractional time-derivative, reformulated as a local Neumann extension problem in the half-space \mathbb{R}^4_+ with boundary condition -\partial_{x_4}u = u^2 on \mathbb{R}^3. The authors construct finite-time Type II blow-up solutions exhibiting a multi-bubble profile by employing a parabolic inner-outer gluing scheme around Aubin–Talenti bubbles, deriving a precise blow-up rate \mu_j(t) ~ |\log 2T| (T-t) / |\log(T-t)|^2 and localization ξ_j(t) → q_j. The analysis hinges on detailed linear theories for the outer heat equation with Neumann boundary and for the inner bubble dynamics (mode 0 and higher modes), coupled with a nonlocal reduced dynamics for the scaling parameter and a fixed-point argument to close the construction. The results provide explicit asymptotics, including a sign condition on the boundary data Z_0^*(q_j,0) < 0, and extend the landscape of known blow-up phenomena to nonlocal parabolic models with Neumann boundary conditions. The methods pave the way for multi-bubble blow-up constructions in related nonlocal parabolic equations and highlight the effectiveness of the inner-outer gluing framework in non-symmetric geometries.
Abstract
We consider the nonlinear heat equations with Neumann boundary conditions $$ \begin{cases} u_{t}=Δu & \text{in}\ \mathbb{R}_{+}^{4} \times(0, T) ,\\ -\frac{d u}{d x_{4}}(\tilde{x}, 0, t) \ =u^2(\tilde{x}, 0, t)& \text{in}\ \mathbb{R}^{3} \times(0, T). \end{cases} $$ 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 $$u(x,t)=\sum_{j=1}^{k}U_{μ_{j}(t),ξ_{j}(t)}(x)+Z_0^*(x)+o(1)\quad \text{as}~ t\nearrow T,$$ where $$U_{μ_{j}(t),ξ_{j}(t)}(x):=μ_{j}^{-1}(t) U\left(\frac{x-ξ_{j}(t)}{μ_{j}(t)}\right),~ x\in \mathbb{R}_{+}^{4},$$ 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 $(-Δ)^{\frac{1}{2}} \widetilde{U} = \widetilde{U}^{2}$ in $\mathbb{R}^{3}$. For some constants $β_{j}>0$, the scaling parameters $μ{j}(t)$ and the translation parameters $ξ_{j}(t)$ satisfy $$μ_{j}(t)=β_{j}\frac{|\log 2T|(T-t)}{|\log(T-t)|^{2}}(1 + o(1)) \to 0,~ξ_{j}(t)\to (q_{j},0)\quad \text{as} ~t\nearrow T.$$