The Propetries Of Solutions For Higher-order Nonlinear Parabolic Equations And Their Numerical Solutions

As mathematics models, higher-order nonlinear parabolic type equations describe many phenomenons which exist in Physics, Chemistry, Informatics, Bioscience, Geo-science, Environmental science, Space science and so on. Higher-order nonlinear parabol-ic type equations is an important part of Nonlinear science.In this paper, we consider the properties of solutions and numerical solutions for three types of higher order nonlinear parabolic equation with extensive physical back-ground.Firstly, we consider the higher order nonlinear parabolic equation describing thin-film epitaxial growth whose diffusion coefficient dependents on the unknown function Using Leyar-Schauder fixed point theorem, the qualities of Campanato space and a prior estimates, we get the existence and uniqueness of global solutions and the existence of classical solutions for IBVP (1).Secondly, we study the long time behavior of solutions for the higher order non-linear parabolic equation describing thin-film epitaxial growth and the higher order nonlinear parabolic equation describing process growing of a crys-tal surface Using the theorem of existence of global attractor in [1,2], we obtain the existence of global attractor for equation (2) in fractal dimension space Hk(0≤k<5) and the existence of global attractor for equation (3) in fractal dimension space Hk(0≤k<+∞). Noticing that the value of p is not only integer, but also fraction in (2), when we consider the a prior estimates for equation (2), if differential many times for term div(|(?)u|p-2(?)u), it maybe exists negative exponential (the a priori estimates of this kind of situation are hard to obtain). So, we have to obtain the existence of global attractor for (2) in space Hk(0≤k<5). On the other hand, there is no this kind of situation for equation (3), we can use iteration method get the existence of global attractor for equation (3) in space Hk(0≤k<+∞).Thirdly, we consider the numerical solutions for the higher order nonlinear parabol-ic equations describing thin-film epitaxial growth and process growing of a crystal sur-face. Using finite element method,[3] studied the Cahn-Hilliard equation Basing on the finite element projection approximation of a biharmonic problem, he ob-tain the optimal order L2-norm error estimate for equation4. In this paper, we consider the finite element method for equation (2) and equation (3). We also use the finite ele-ment projection approximation of a biharmonic problem, obtain the optimal order L2-norm error estimate, get the numerical analysis. Moreover, in [3], the nonlinear term of equation is (?)2/(?)x2φ(u). when establishing the approximation form, we can integration by parts two times (under sufficiently boundary value conditions) and the a prior estimates, that is where Sh(k) is a finite element space which constituted by the k≥3piecewise polyno-mial. Then, using the finite element analysis method for biharmonic equation [4], we can obtain the result. But the nonlinear terms of the two equations considered by us are different from [3]. When we establish the approximation form, we can only integration by parts one time (under sufficiently boundary value conditions), using the result of a prior estimate, we get Because there exist the derivative term, by the finite element analysis method for bihar-monic equation [4], the error precision of space will decline in one order.The modified Swift-Hohenberg equation is also a higher order nonlinear parabolic equation with a derivative term. Using Fourier spectral method, we study the numerical solutions for equation (5). We establish the semi-discrete form and full-discrete form, get the error estimates. We can also meet the same problem when we analyse the equation. But, the derivative term is a quadratic term, using sufficient integration by parts and the a prior estimates, we can avoid the decline of error precision of space.