Asymptotic expansion of solutions to the drift-diffusion equation with large initial data
The Cauchy problem to the drift-diffusion equation is considered. \[ \begin{aligned} &\frac{\partial u}{\partial t}+\nabla\cdot(u\nabla \psi)-\Delta u=0, \quad -\Delta\psi=-ku,\quad x\in\mathbb R^n,\;t>0\\ &u(x,0)=u_0(x),\quad x\in\mathbb R^n, \end{aligned} \] where \(n\geq 3\), \(k=\pm 1\), \(u(x,t)\) is the unknown function. This equation with \(k=1\) is known as the model of the plasma dynamics. When \(k=-1\) the equation stands for the motion of the gravitational interaction particles in astronomy. The large time behavior of the solution to the problem is shown in the paper. This behavior is different for odd an even \(n\). If \(k=1\) the initial condition \(u_0\) can be large, but for \(k=-1\) \(u_0\) must be small in \(L_p\) norm.
scientific article; zbMATH DE number 5731672
scientific article