Well-posedness of the Cauchy problem for the fractional power dissipative equations

Abstract

This paper studies the Cauchy problem for the nonlinear fractional power dissipative equation $u_t+(-△{)}^{\alpha }u=F(u)$ for initial data in the Lebesgue space $L^r(\mr^n)$ with $\ds r\ge r_d\triangleq{nb}/({2α-d})$ or the homogeneous Besov space $\ds\dot{B}^{-σ}_{p,\infty}(\mr^n)$ with $\dsσ=(2α-d)/b-n/p$ and $1\le p\le \infty$, where $\alpha >0$, $F(u)=f(u)$ or $Q(D)f(u)$ with $Q(D)$ being a homogeneous pseudo-differential operator of order $d\in [0,2\alpha )$ and $f(u)$ is a function of $u$ which behaves like $|u{|}^{b}u$ with $b>0$.

30pages

Cited by in corpus (9)

• Global well-posedness of the critical Burgers equation in critical Besov spaces
• On well-posedness of the Cauchy problem for MHD system in Besov spaces
• On convergence of solutions of fractal Burgers equation toward rarefaction waves
• Well-posedness and regularity of generalized Navier-Stokes equations in some Critical $Q-$spaces
• Asymptotic behavior for dissipative Korteweg-de Vrie equations
• Global Existence of Solutions to the 2D subcritical dissipative Quasi-Geostrophic equation and persistency of the initial regularity
• Well-posedness and ill-posedness of the 3D generalized Navier-Stokes equations in Triebel-Lizorkin spaces
• A Tracing of the Fractional Temperature Field
• Decay of solutions to the three-dimensional generalized Navier-Stokes equations