In this article, we consider the nonlinear heat equation u(t) = Deltau + \u\(p-1)u on Omega x R+ with some boundary conditions and the initial condition u(x, 0) = u(0)(x) is an element of L-1(Omega), where Omega subset of R-n and p > 1. For the one dimensional case, it is well known that for p < 3 this problem has a local solution for any initial condition u(0) is an element of L-1(Omega). But the existence and uniqueness of a local solution in L-1 for the critical exponent p = 3 was wide open and this work is to answer this open question. First, we prove that for the Cauchy problem there is no local solution in L-1 for some u(0) is an element of L-1. Then using the nonlocal existence of Cauchy problem by a cutoff function argument, we prove the non-local existence of a solution for the Dirichlet problem which answers this open question. Moreover, we generalize the nonlocal existence result for n-dimensional case with the critical exponent p = 1 + 2/n. More general nonlinearity is also considered for Dirichlet boundary value problems. Finally, we prove the same result for the mixed boundary condition with the same initial data u(0).