A Summary For The Development Of A Nonlinear Fourth-Order Parabolic Equation

The development of a nonlinear fourth-order parabolic equation is summarized in this paper, the origin and the physical background of the equations are referred to and the difficulty when studying the problem, and the research results up to now are also mentioned in this paper. The fourth-order parabolic equation was first derived in quantum semiconductor modelling. And there are only few results until now. In this paper, we emphasize on the result about the equation u_t = - (u(logu)_xx)_xx with weak initial conditions. The main way to solve this kind of problem is to employ an exponential transformation, for the new variable we get a new equation. Then use the semidiscretization of time, and let the length of the time interval tends to 0, take the limit, we get the existence of a solution. For higher dimensional case (2 or 3 dimensions), giving suitable initial and boundary conditions, there are also some results. And it is an effective way to solve the Cauchy problem for the equation with semigroup theory. For good initial conditions(eg. H¹(Ω) functions)it is proved this equation has a local classical solution[4], for initial conditions good (the derivative is small) enough, the solution is even global. For the long-time behavior ,that is to say whether the solutions tend to the limit solution and the convergence speed when the time variable t tend to ∞, suitable entropy is generally evaluated first, and then we get the convergence speed for the aimed function through the connection of the aimed function and the entropy.