ASYMPTOTICS FOR FRACTIONAL NONLINEAR HEAT EQUATIONS

The Cauchy problem is studied for the nonlinear equations with fractional power of the negative Laplacian ut + (−Δ)α/2u + u1+σ =0, x∈ Rn, t> 0, u(0, x)=u0(x), x∈ Rn , where α ∈ (0, 2), with critical σ = α/n and sub-critical σ ∈ (0, α/n) powers of the nonlinearity. Let u0 ∈L1,a ∩L∞∩C, u0(x) 0 in Rn , θ = Rn u0(x) dx > 0. The case of not small initial data is of interest. It is proved that the Cauchy problem has a unique global solution u ∈ C([0,∞);L∞∩ L1,a ∩ C) and the large time asymptotics are obtained.