带有非线性阻尼项的广义Hartree方程的整体适定性

Global Well-posedness for Generalized Hartree Equation with a Nonlinear Damping

本文研究带有非线性阻尼项的广义Hartree方程的整体适定性:{i■_(t)u+Δu+ia|u|^(q-2)u=±(Iα*|u|^(p))|u|^(p-2)u,(t,x)∈R^(+)×R^(d),u(0)=u_(0),x∈R^(d),其中d≥3,a>0,0<α<d.当2≤p<d+α/d-2且2≤q<2d/d-2时,我们得到了该方程的局部适定性;当p=1+2+α/d,2≤q<2d/d-2且‖u0‖L^(2)≤‖Q‖L^(2)时,我们证明了解是整体存在的;当p=1+2+α/d且q=2+4/d时,我们发现阻尼项可以阻止解的爆破,并进一步研究了解的散射问题.

This article studies global well-posedness of generalized Hartree equation with a nonlinear damping{i■_(t)u+Δu+ia|u|^(q-2)u=±(Iα*|u|^(p))|u|^(p-2)u,(t,x)∈R^(+)×R^(d),u(0)=u_(0),x∈R^(d),其中d≥3,a>0,0<α<d,where d≥3.α>0.0<α<d.whe n2≤p<d+a/d-2and 2≤q<2d/d-2,we get the local well-posedness for this equation;When p=1+2+a/d,2≤q<2d/d-2 and|‖u0‖L^(2)≤‖Q‖L^(2),we prove the solution is global;When p=1+2+a/d and q=2+4/d,we find that the damping prevents blow-up,and further study the scattering problem.