Singular Limit as q→∞ for a Doubly Nonlinear Cauchy Problem with Absorption

In this paper, we study the singular limit q→∞ for non-negative weak solutions of a doubly nonlinear parabolic Cauchy problem with absorption. Under the assumptions m>1, m (p−1) >1 and q>p (m+2), we first prove that the family u (q) is precompact, uniformly on compact subsets of RN× (0, ∞), and that the whole family converges to the unique bounded weak solution of the homogeneous problem with initial datum v0 (x) =minu0 (x), 1. We then show that the limit operator Tt is given by the composition of the homogeneous semigroup with the truncation u0↦minu0, 1; it forms an order-preserving semigroup, satisfies the L1-contraction property, and exhibits an initial layer precisely characterized by the excess mass (u0−1) +. Finally, we compare the singular limit with the homogeneous solution and give a sharp description of the corresponding convergence defect.