Nonuniqueness in the quenching problem

Abstract

This paper deals with nonnegative solutions of for

$$u_t = \Delta u - u^{-q} \chi_{\{u > 0\}} \qquad {\rm in}\,\Omega \times (0,\infty) \qquad \qquad ({\rm Q})$$

with $q \in (0, 1)$ and prescribed continuous Dirichlet data B = B(x) on ∂Ω. It is proved that for n ≤ 6 there is a critical parameter$q_c \in (0, 1)$ with the following property: If q >  q _c then there exist at least two continuous weak solutions emanating from some explicitly known stationary solution w: one that coincides with w and another one that satisfies u ≥ w but $u \not\equiv w$ . For n ≤ 6 and q ≤ q _c (or n ≥ 7), however, such a second solution above w is impossible. Moreover, it is shown that for n ≤ 6, q >  q _c and any sufficiently small nonnegative boundary data B there exist initial values admitting at least two continuous weak solutions of (Q). The final result asserts that for any n and q nonuniqueness for (Q) holds at least for some boundary and initial data.