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Classification theory on the existence and non-existence of local in timesolutions for initial value problems of nonlinear heat equations areinvestigated. Without assuming a concrete growth rate on a nonlinear term, wereveal the threshold integrability of initial data which classify existence andnonexistence of solutions via a quasi-scaling and its invariant integral.Typical nonlinear terms, for instance polynomial type, exponential type and itssum, product and composition, can be treated as applications.

Existence and nonexistence of solutions for the heat equation with a superlinear source term