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With Q ⊂ R N a bounded open connected set, the initial value problem for the nonhomogeneous nonlinear heat equation ∂u/∂t - ΔM = f(x, t, u) with zero boundary conditions and zero initial condition is solved in a generalized sense in the region Ω × (0, T) under new one-sided conditions on f(x, t, s). In particular, this result greatly improves on the well-known one-sided result of Brezis and Nirenberg, [2, p. 302]. The techniques used here are completely different from those employed by the last mentioned authors and carry over even to the case when the space operator may be singular elliptic and quasilinear in the lower order terms. So the main result is presented in this singular quasilinear context.