A Very Singular Solution of the Heat Equation with Absorption

Consider the Cauchy problem ut -du + u p ----0 on RN• (0, oo) (I.1) u > 0 on RN• (0, oo) (1.2) u(X, O) = c ~(x) on R, N, (1.3) where N _--> 1, c > 0 is a constant and ~(x) denotes the Dirac mass at the origin. A result of BREZlS and FRIEDMAN [6] asserts that if 1 < p < (N + 2)/N, then for every c > 0 there exists a unique 1 solution uc of (1.1)-(1.3). When p >= (N + 2)IN there is no solution of (1.1)-(1.3) and in fact any solution u of (1.1) such that u ~ 0 on RN• oo) and lim f u(x, t) Z(x) dx = 0 VT, ~ Co~ N \ (0}) t~,o RN