Global existence for some slightly super-linear parabolic equations with measure data

In this work we study the global existence of a solution to some parabolic problems whose model is ut − u = g(u)+μ, (x, t) ∈ Ω ×(0,∞), u(x, t) = 0, (x, t) ∈ ∂Ω ×(0,∞), u(x, 0) = u0(x), x ∈ Ω, (1) where Ω ⊂ RN is a bounded domain, u0 ∈ L1(Ω), μ is a finite Radon measure in Ω × (0,∞) and g is a real continuous function, slightly superlinear at infinity (“slightly” in the sense that 1/g is not integrable at ∞). One of the main tools is a new logarithmic Sobolev inequality. We also prove some uniqueness results.