Quasilinear parabolic equations with localized reaction

Abstract

In this paper, we study a nonnegative blow-up solution of the Dirichlet problem for a quasilinear parabolic equation $(u^{\alpha})_t=\Delta u +f(u) + g(u(x_0(t),t))$ in $B(R)$, where $B(R)=\{ x \in \mathbf{R^N}\,;\, |x| < R\}$, $0 < \alpha \le 1$, $x_0(t)\in C^{\infty}([0,\infty) ;B(R))$ satisfies $x_0(t)\not = 0$, and $f(\xi)$ and $g(\xi)$ satisfy some blow-up conditions. We consider radial blow-up solutions $u(r,t)$ ($r=|x|$) which are non-increasing in $r$, and give the classification between total blow-up and single point blow-up according to the growth orders of $f$ and $g$. Especially in the case $\alpha =1$ we completely classify blow-up phenomena except for $f\sim g$ as follows. (I) When $g=o(f)$, any blow-up solution blows up only at the origin (single point blow-up); (II) When $f=o(g)$, (i) a single point blow-up solution exists (ii) there exists an initial data such that the solution blows up in the whole domain $B(R)$ (total blow-up) (iii) there are no other blow-up phenomena except total blow-up and single point blow-up.