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Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains. (English) Zbl 1218.35052
The problem
\begin{alignedat}{3} u_t-\Delta u&=|\nabla u|^p,&\quad &x\in\Omega,&\quad &t>0,\tag{1}\\ u(x,t)&=0,&\quad &x\in\partial\Omega,&\quad &t>0,\tag{2}\\ u(x,0)&=u_0(x), &\quad &x\in\Omega,&&{}\tag{3} \end{alignedat}
where $$u_0\in X_+=\{v\in C^1(\overline\Omega$$); $$v\geq 0$$, $$v|_{\partial\Omega} =0\}$$, is considered. This problem admits a unique maximal, nonnegative classical solution $$u\in C^{2,1}(\overline \Omega \times (0,T))\cap C^{1,0}(\overline\Omega \times [0,T))$$, where $$T=T(u_0)$$ is the maximal existence time and $$\|u(t)\|_\infty \leq \|u_0\|_\infty$$, $$0<t<T$$, by the maximum principle. Sine (1)–(3) is well-posed in $$X_+$$, it follows that, if $$T<\infty$$, then
$\lim_{t \to T}\|\nabla u(t)\|_\infty = \infty .$
This phenomenon of $$\nabla u$$ blowing up with $$u$$ remaining uniformly bounded is known as gradient blow-up. The question of the location of gradient blow-up points within the boundary for problem (1)–(3) with $$n \geq 2$$ has not been addressed so far. The gradient blow-up set of $$u$$ is defined by
$\text{GBUS}(u_0)=\{ x_0\in \partial\Omega;\;\nabla u\text{ is unbounded in }(\overline\Omega\cap B_\eta (x_0))\times (T-\eta ,T) \text{ for any } \eta >0 \}.$
Note that by definition, $$\text{GBUS}(u_0)$$ is compact. The main goal of this paper is to show that under some assumptions on $$\Omega \subset \mathbb R^2$$ and $$u_0$$, the gradient blow-up set $$\text{GBUS}(u_0)$$ contains only one point. A possible physical interpretation is that the surface tension (diffusion) forces the steep region to become more and more concentrated near a single boundary point.

##### MSC:
 35B44 Blow-up in context of PDEs 35K58 Semilinear parabolic equations 82C24 Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics
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