On some nonlinear dissipative equations with sub-critical nonlinearities

We study the Cauchy problem for the nonlinear dissipative equations (1) {∂_tu + α (-Δ)^ρ/2 u + β|u| ^σ u + γ|u|^x u = 0, x ∈ Rⁿ,t > 0, u(0,x) = u₀(x), x ∈ Rⁿ, where α,β,γ ∈ C, Re a > 0, ρ > 0, x > σ > 0. We are interested in the critical case, σ = ρ/n and sub critical cases 0 < σ < ρ/n. We assume that the initial data u₀ are sufficiently small hi a suiatble norm, |∫u₀ (x) dx| > 0 and Reβδ(α,p,σ) > 0, where δ(αρ, σ) = ∫|G(x)|σ(x)dx and G (x) = ℱ^-1e- ^α|ξ|ρ. In the sub critical case we assume that CT is close to ρ/n. Then we prove global existence in time of solutions to the Cauchy problem (1) satisfying the time decay estimate δ(α,ρ, σ) ∫|G(x)^σ G(x)dx ||u(t)||_{L ∞} ≤{(C(1 +t)^-1/σ(log(2+t)^-1/σif σ = ρ/n, C (1+t)-1/σif σ ∈(0, ρ/n).