Isolated singularities of nonlinear parabolic inequalities

Abstract

We study C ^2,1 nonnegative solutions u(x,t) of the nonlinear parabolic inequalities

$$ 0\le u_t-\Delta u \le u^\lambda$$

in a punctured neighborhood of the origin in ${\bf R}^n \times [0,\infty)$, when $n\ge 1$ and $\lambda > 0$. We show that a necessary and sufficient condition on λ for such solutions u to satisfy an a priori bound near the origin is $\lambda\le \frac{n+2}n$, and in this case, the a priori bound on u is

$$ u(x,t) = O(t^{-n/2}) \quad \text{as}\; (x,t)\to (0,0),\; t > 0.$$

This a priori bound for u can be improved by imposing an upper bound on the initial condition of u.