The heat equation with singular nonlinearity and singular initial data

Abstract

We study the existence, uniqueness and regularity of positive solutions of the parabolic equation u t − Δ u = a ( x ) u q + b ( x ) u p in a bounded domain and with Dirichlet's condition on the boundary. We consider here a ∈ L α ( Ω ) , b ∈ L β ( Ω ) and 0 < q ⩽ 1 < p . The initial data u ( 0 ) = u 0 is considered in the space L r ( Ω ) , r ⩾ 1 . In the main result ( 0 < q < 1 ), we assume a , b ⩾ 0 a.e. in Ω and we assume that u 0 ⩾ γ d Ω for some γ > 0 . We find a unique solution in the space C ( [ 0 , T ] , L r ( Ω ) ) ∩ L loc ∞ ( ( 0 , T ) , L ∞ ( Ω ) ) .