A complete characterisation of local existence for semilinear heat equations in Lebesgue spaces

We consider the scalar semilinear heat equation , where is continuous and non-decreasing but need not be convex. We completely characterise those functions f for which the equation has a local solution bounded in for all non-negative initial data , when is a bounded domain with Dirichlet boundary conditions. For this holds if and only if ; and for if and only if , where . This shows for the first time that the model nonlinearity is truly the ‘boundary case’ when , but that this is not true for