Single-point gradient blow-up on the boundary for diffusive Hamilton-Jacobi equations in planar domains
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.
scientific article; zbMATH DE number 5852643
scientific article