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 isequation(1){ut−Δu=g(u)+μ,(x,t)∈Ω×(0,∞),u(x,t)=0,(x,t)∈∂Ω×(0,∞),u(x,0)=u0(x),x∈Ω, where Ω⊂RNΩ⊂RN is a bounded domain, u0∈L1(Ω)u0∈L1(Ω), μ   is a finite Radon measure in Ω×(0,∞)Ω×(0,∞) and g is a real continuous function, slightly superlinear at infinity   (“slightly” in the sense that 1/g1/g is not integrable at ∞). One of the main tools is a new logarithmic Sobolev inequality. We also prove some uniqueness results.