LAUSR

Abstract We study the existence of solutions of the semilinear parabolic equation ∂ t u − Δ u = u p + Λ in D ′ ( R N × I ) . Here N ≥ 2 , p > 1 , ( N − 2 ) p N and I ⊂ R is an open interval. It is assumed that Λ ∈ D ′ ( R N × I ) is supported on { ( ξ ( t ) , t ) ∈ R N + 1 ; t ∈ I } for some ξ ∈ C 1 / 2 ( I ‾ ; R N ) and is given by Λ ( φ ) = ∫ I φ ( ξ ( t ) , t ) d μ ( t ) , where μ is a Radon measure on I satisfying μ ( ( a , b ) ) ≤ ϕ ( b − a ) ( a , b ∈ I ) for some positive function ϕ ( η ) . In the case where I is bounded, we give conditions on the behavior of ϕ ( η ) near η = 0 for the existence and nonexistence of solutions. The existence for I = R is also shown under additional conditions on p and the behavior of ϕ ( η ) near η = ∞ . Moreover, the singularity of solutions at x = ξ ( t ) is examined.