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

Abstract

Abstract We consider nonnegative solutions of a semilinear heat equation u t − Δ u = u p in R N ( N ≥ 3 ) with p = N / ( N − 2 ) and a nonnegative initial data u 0 ∈ L N / ( N − 2 ) ( R N ) which has a singularity at ξ 0 ∈ R N . We prove that there exists u 0 such that, for any ξ ∈ C α ( [ 0 , ∞ ) ; R N ) with α ∈ ( 1 / 2 , 1 ] and ξ ( 0 ) = ξ 0 , the problem admits a nonnegative solution u ξ ∈ C ( [ 0 , T ξ ] ; L N / ( N − 2 ) ( R N ) ) for some T ξ with an explicit singular leading term for each t ∈ [ 0 , T ξ ] 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.