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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

u t -Δu=|u| p ,xΩ,t>0,u(x,t)=0,xΩ,t>0,u(x,0)=u 0 (x),xΩ,

where u 0 X + ={vC 1 (Ω ¯); v0, v| Ω =0}, is considered. This problem admits a unique maximal, nonnegative classical solution uC 2,1 (Ω ¯×(0,T))C 1,0 (Ω ¯×[0,T)), where T=T(u 0 ) is the maximal existence time and u(t) u 0 , 0<t<T, by the maximum principle. Sine (1)–(3) is well-posed in X + , it follows that, if T<, then

lim tT u(t) =·

This phenomenon of 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 n2 has not been addressed so far. The gradient blow-up set of u is defined by

GBUS(u 0 )={x 0 Ω;uisunboundedin(Ω ¯B η (x 0 ))×(T-η,T)foranyη>0}·

Note that by definition, GBUS(u 0 ) is compact. The main goal of this paper is to show that under some assumptions on Ω 2 and u 0 , the gradient blow-up set 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.

35B44Blow-up (PDE)
35K58Semilinear parabolic equations
82C24Interface problems (dynamic and non-equilibrium); diffusion-limited aggregation
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