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

Source abstract

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)=min{u0(x),1}. We then show that the limit operator Tt is given by the composition of the homogeneous semigroup with the truncation u0↦min{u0,1}; 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.