Well-posedness And Asymptotic Properties Of Solutions To Two Fourth Order Nonlinear Parabolic Partial Differential Equations

With the development of modern mathematics and computer science, both the theory and applications of partial differential equation (PDE) have made great improvements. Many higher order partial differential equations were derived from the area of physics, geometry, mechanics and engineering, etc. However, higher order equations are different from second order equations mainly in the lack of maximum principles, and indeed demand some different theoretic tech-niques in order to obtain a priori estimate and the nonnegativity or positivity of solutions. At the same time, such equations describing and explaining natural phenomena are usually nonlinear, even degenerate or singular, which make the study of higher order PDE more complicated and difficult. Still, the study of higher order partial differential equations has attracted more and more scholars’attention.In this thesis, we discuss the well-posedness and asymptotic properties of two fourth order nonlinear PDE models. More precisely, we study the finite speed of propagation property and long time behavior of strong solutions to a generalized thin film equation, and furthermore, the global existence, uniqueness and asymptotic behavior of classical solutions to a fourth order equation related to image processing. The dissertation, which can be divided into two parts, is organized as follows.In the first part, we consider the following generalized thin film equation ut+[un(uxxx-cux+b)]x=0, x∈Ω, t>0under the periodic boundary conditions, where Ω(?) R is a bounded interval, b, c≥0and the real number n is positive. We first consider the existence of nonnegative solutions to the initial boundary value problem and establish some essential estimates. Secondly, based on combined use of local entropy estimate, local energy estimate and the suitable extensions of Stampacchia’s lemma to systems, we obtain the finite speed of propagation property of strong solutions in the case of weak slippage2≤n<3. Finally, the long time behavior of positive classical solutions to the initial boundary value problem for the case of n=1is investigated. By establishing the entropy dissipation inequality, we show that the solution decays to the mean value uniformly in x, in an order like t-1/4for long times. It is worth mentioning that our conclusions improve and extend the previous results.The second part of the thesis is mainly devoted to the following fourth order parabolic equation for noise removal ut+(g(u)uxx)xx=0, x∈Ω,t>0,(0.2) where g(u)=u-n, n>0, Ω is a bounded interval. First of all, the initial boundary value prob-lem with the homogeneous Neumann and no-flux boundary conditions is considered. Applying the algebraic approach, we first derive some entropy estimates which are the key ingredients for our results. Moreover, using the approximation method combined with the entropy esti-mate, we prove that the initial boundary problem admits a unique global classical solution, and furthermore, the solution converges to its mean value as the time tends to infinity. Secondly, we mention the fact that our approach can also be used to handle the corresponding Dirichlet boundary value problem, the similar results on the well-posedness and the long time behavior of solutions are shown. Lastly, we talk about the higher dimensional case of the model (0.2) and establish the entropy dissipation estimates for the radially symmetric solutions.