Existence of solutions with moving singularities for a semilinear heat equation with a critical exponent

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We consider nonnegative solutions of a semilinear heat equation ut−Δu=up in RN (N≥3) with p=N/(N−2) and a nonnegative initial data u0∈LN/(N−2)(RN) which has a singularity at ξ0∈RN. We prove that there exists u0 such that, for any ξ∈Cα([0,∞);RN) with α∈(1/2,1] and ξ(0)=ξ0, the problem admits a nonnegative solution uξ∈C([0,Tξ];LN/(N−2)(RN)) for some Tξ with an explicit singular leading term for each t∈[0,Tξ] as x→ξ(t). Our result refines known counter examples for the uniqueness of the doubly critical case in view of pointwise behavior and complements known sufficient conditions on p for the existence of solutions with moving singularities.