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We consider the scalar semilinear heat equation u t − Δ u = 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 u 0 ∈ L q ( Ω ) , when Ω ⊂ R d is a bounded domain with Dirichlet boundary conditions. For q ∈ ( 1 , ∞ ) this holds if and only if lim sup s → ∞ s − ( 1 + 2 q / d ) f ( s ) < ∞ ; and for q = 1 if and only if ∫ 1 ∞ s − ( 1 + 2 / d ) F ( s ) d s < ∞ , where F ( s ) = sup 1 ≤ t ≤ s ⁡ f ( t ) / t . This shows for the first time that the model nonlinearity f ( u ) = u 1 + 2 q / 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 lim sup s → 0 f ( s ) / s < ∞ . [ABSTRACT FROM AUTHOR]

Abstract: We establish non-existence results for the Cauchy problem of some semilinear heat equations with non-negative initial data and locally Lipschitz, non-negative source term f. Global (in time) solutions of the scalar ODE exist for if and only if the Osgood-type condition holds; by comparison this ensures the existence of global classical solutions of for bounded initial data . It is natural to ask whether the Osgood condition is sufficient to ensure that the problem still admits global solutions if the initial data is in for some . Here we answer this question in the negative, and in fact show that there are initial conditions for which there exists no local solution in for . [Copyright &y& Elsevier]