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We prove that as m !1 the solutions u(m) of ut = u m u p , (x;t) 2 Rn (0;T), T> 0, m> 1, p> 1, u 0, u(x; 0) = f(x) 2 L1(Rn) \ L1(Rn), converges in L1 loc(R n (0;T)) to the solution of the ODE vt = vp, v(x; 0) = g(x), where g 2 L1(Rn), 0 g 1, satises g eg = f in D0(Rn) for some function eg 2 L1 loc(R n), eg 0, satisfying eg(x) = 0 whenever g(x) < 1 for a.e. x 2 Rn, R E egdx CjEj2=n for n 3a nd R E jregjdx CjEj1=2 for n =2 , where C> 0i s ac onstant and Eis any measurable subset of Rn. In this paper we will show that as m !1 the solutionsu = u(m) of the equation ( ut = u m up;u 0; in Rn (0;T ); u (x; 0) =f(x) 0;x2R n ;f 2 L 1 ( R n ) \ L 1 ( R n ) ; (0.1)

Singular limit of solutions of the porous medium equation with absorption