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 (34)

• Global Regularity of 2D Generalized MHD Equations with Magnetic Diffusion
• 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
• Continuous dependence estimates for nonlinear fractional convection-diffusion equations
• Non-uniqueness of weak solutions for the fractal Burgers equation
• Forward self-similar solutions of the fractional Navier-Stokes Equations
• A Blow-up Dichotomy for Semilinear Fractional Heat Equations
• An inverse problem for a fractional diffusion equation with fractional power type nonlinearities
• Global Regularity of weak solutions to the generalized Leray equations and its applications
• Well-posedness and regularity of generalized Navier-Stokes equations in some Critical $Q-$spaces
• Global solutions to Stokes-Magneto equations with fractional dissipations
• On the dimension of divergence sets of Schrödinger equation with complex time
• Hölder regularity for the time fractional Schrödinger equation
• Global well-posedness, dissipation and blow up for semilinear heat equations in energy spaces associated with self-adjoint operators
• Asymptotic behavior for dissipative Korteweg-de Vrie equations
• Nonuniqueness of Leray-Hopf solutions to the forced fractional Navier-Stokes Equations in three dimensions, up to the J. L. Lions exponent
• Global existence and spatial analyticity for a nonlocal flux with fractional diffusion
• Global existence of uniformly locally energy solutions for the incompressible fractional Navier-Stokes equations
• Global well-posedness for 2D fractional inhomogeneous Navier-Stokes equations with rough density
• Local well-posedness of semilinear space-time fractional Schrödinger equation
• Deterministic ill-posedness and probabilistic well-posedness of the viscous nonlinear wave equation describing fluid-structure interaction
• Global Existence of Solutions to the 2D subcritical dissipative Quasi-Geostrophic equation and persistency of the initial regularity
• Time-decay estimates for linearized two-phase Navier-Stokes equations with surface tension and gravity
• Boundedness of differential transforms for heat semigroups generated by fractional Laplacian
• Decay of solutions to the three-dimensional generalized Navier-Stokes equations
• Blow up of fractional Schrödinger equations on manifolds with nonnegative Ricci curvature
• Fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations
• Bochner's Subordionation and Fractional Caloric Smoothing in Besov and Triebel--Lizorkin Spaces
• A Tracing of the Fractional Temperature Field
• Global existence for the nonlinear fractional Schrödinger equation with fractional dissipation
• Arrest of blowup for the 3-D semi-relativistic Schrödinger-Poisson system with pseudo-relativistic diffusion
• Mixed Lebesgue space norm Strichartz type estimation for solution of inhomogeneous parabolic equation, with constants evaluation
• Well-posedness and ill-posedness of the 3D generalized Navier-Stokes equations in Triebel-Lizorkin spaces