Let $u = (u_1,u_2,u_3)$ be a divergence free vector, $P$ a skew symmetric 3 by 3 matrix with determinant zero, $dx = (dx_1,dx_2,dx_3)$ and $g$ a scalar field. Consider the reaction diffusion equation:
$[ u_t - \Delta_x u -Pu + \nabla_x(||u||^2)].dx = \nabla_x g.dx,$
$u(x,0)= u_0(x).$
I want to study the existence and asymptotic behavior of the solution.
I wanted trying the energy method but i don't know how to define the energy function in this case. Any suggestions to this or other approach will be appreciated. Thanks