On the Blow-Up Set for Non-Newtonian Equation with a Nonlinear Boundary Condition

Abstract

We identify the blow-up set of solutions to the problem $u_t=\left(\right|u_x{|}^{p-2}{u}_x{)}_x$, $x>0$, $t\in (0,T)$, $-\left|u_x{|}^{p-2}{u}_x\right(0,t)=u^{p-1}(0,t)$, $t\in (0,T)$, and $u(x,0)=u_0\left(x\right)\ge 0$, $x>0$, where $p>2$. We obtain that the blow up set $B\left(u\right)$ satisfies $[0,p(p-1)/(p-2\left)\right)\subset B\left(u\right)\subset [0,p(p-1)/(p-2\left)\right]$. The proof is based on the analysis of the asymptotic behavior of self-similar representation and on the comparison methods.