Nonuniqueness in the quenching problem

This paper deals with nonnegative solutions of for [GRAPHICS] with q epsilon (0, 1) and prescribed continuous Dirichlet data B = B(x) on partial derivative Omega. It is proved that for n <= 6 there is a critical parameter q(c) epsilon [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 equivalent to w. For n <= 6 and q <= qc (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.