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

We consider the scalar semilinear heat equation u_t−Δu=f(u), where f:[0,∞)→[0,∞) 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 L^q(Ω) for all non-negative initial data u₀∈L^q(Ω), when Ω⊂R^d is a bounded domain with Dirichlet boundary conditions. For q∈(1,∞) this holds if and only if limsup_s→∞s^−(1+2q/d)f(s)<∞; and for q=1 if and only if ∫₁^∞s^−(1+2/d)F(s)ds<∞, where F(s)=sup_1≤t≤s⁡f(t)/t. This shows for the first time that the model nonlinearity f(u)=u^1+2q/d is truly the ‘boundary case’ when q∈(1,∞), but that this is not true for q=1. The same characterisations hold for the equation posed on the whole space R^d provided that limsup_s→0f(s)/s<∞.