## Abstract

We study the Cauchy problem for the nonlinear dissipative equations (1) {∂_{t}u + α (-Δ)^{ρ/2} u + β|u| ^{σ} u + γ|u|^{x} u = 0, x ∈ R^{n},t > 0, u(0,x) = u_{0}(x), x ∈ R^{n}, 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 u_{0} are sufficiently small hi a suiatble norm, |∫u_{0} (x) dx| > 0 and Reβδ(α,p,σ) > 0, where δ(αρ, σ) = ∫|G(x)|σ(x)dx and G (x) = ℱ^{-1}e- ^{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/σ}if σ = ρ/n, C (1+t)-1/σif σ ∈(0, ρ/n).

Original language | English |
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Pages (from-to) | 135-154 |

Number of pages | 20 |

Journal | Taiwanese Journal of Mathematics |

Volume | 8 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2004 |

## Keywords

- Nonlinear dissipative equations
- Sub-critical nonlinearities