Local and global properties of solutions of heat equation with superlinear absorption

Abstract

We study the limit when k →∞ of the solutions of ∂tu-Δu+f(u)=0 in ℝN ×(0,∞) with initial data k, when f is a positive superlinear increasing function. We prove that there exist essentially three types of possible behaviour according to whether f-1 and F-1/2 belong or not to L1(1,∞), where F(t) = sh{phonetic}0t f(s)ds. We use these re-sults for providing a new and more general construction of the initial trace and some uniqueness and nonuniqueness results for solutions with unbounded initial data.