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Zbl 1218.35052
Yuxiang, Li; Souplet, Philippe
Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains. (English)
[J] Commun. Math. Phys. 293, No. 2, 499-517 (2010). ISSN 0010-3616; ISSN 1432-0916/e

The problem $$\alignat3 u_t-\Delta u&=|\nabla u|^p,&\quad &x\in\Omega,&\quad &t>0,\tag1\\ u(x,t)&=0,&\quad &x\in\partial\Omega,&\quad &t>0,\tag2\\ u(x,0)&=u_0(x), &\quad &x\in\Omega,&&{}\tag3 \endalignat$$ where $u_0\in X_+=\{v\in C^1(\overline\Omega$); $v\ge 0$, $v|_{\partial\Omega} =0\}$, 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 $\|u(t)\|_\infty \le \|u_0\|_\infty$, $0<t<T$, 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 $n \ge 2$ 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 $\Omega \subset \Bbb R^2$ 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.
[Vasile Iftode (Bucureşti)]

MSC 2000:: *35B44
35K58
82C24 Interface problems (dynamic and non-equilibrium)

Keywords: diffusive Hamilton-Jacobi equations; gradient blow-up set

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