The Cauchy problem for a semilinear heat equation with singular initial data Bernhard Ruf and Elide Terraneo Dipartimento di Matematica ”F. Enriques” Universit`a di Milano via Saldini, 50, I–20133 MILANO E-mail: ruf@mat.unimi.it, terraneo@mat.unimi.it November 9, 2001 In memory of Brunello Terreni Abstract We review some known results of local existence in the framework of Le- besgue spaces for the nonlinear heat equation with polynomial nonlinearity. Then we consider nonlinearity of exponential growth and we present a new result of local existence in the context of Orlicz spaces. We consider the Cauchy problem for the semilinear heat equation  ∂ t u =∆u + f (u) u(0) = u 0 where u(t, x): R + × R → R, and f ∈C 1 (R, R) is a given function with f (0) = 0. It is well-known that if the initial data u 0 belong to L ∞ (R n ) then there exist T (u 0 ) > 0 and a unique solution u ∈ C ([0,T [, L ∞ (R n )). In this paper we will consider initial data u 0 which do not belong to L ∞ (R n ). At first we will review the known results about this problem; then we will present some new results concerning nonlinearities of exponential growth. The first works in this direction are due to Weissler [24],[25], who mainly con- siders the case of polynomial nonlinearities f (u)= |u| α u, α> 0. He studies the Cauchy problem  ∂ t u =∆u + |u| α u u(0) = u 0 (1) 1