Mathematical Analysis For Several Nonlinear Fourth-order Partial Differential Equations

The paper mainly studies several fourth-order nonlinear partial differential equations,including the existence and regularity for a double degenerate fourth-order parabolic equations,a viscous fourth-order parabolic equations with boundary degeneracy,and the eigenvalue problems for a nonlinear fourth-order elliptic Equations.For the methods,the comparison principle and maximum principle are no longer valid,and so it is necessary to apply energy methods to give the corresponding existence.Chapter 1 is used to mainly explain the research background and research status of the paper,and list the research problems and main results.Chapter 2 studies the existence of solutions for a fourth-order p-Laplacian equations with boundary degenerates.For this,we establish the corresponding regularization problem and semi-discrete elliptic problems.To solve the regularization problem with the non-degenerate fourth-order term coefficients,we mainly use the minimizing functional method and iterative estimates to give the existence and uniqueness.After that,the existence and uniqueness for non-degenerate parabolic problem are obtained by energy estimate and compactness argument.Finally,the existence and regularity of solutions to boundary degeneracy problems are obtained by using the vanishing limit for parameters.Chapter 3 studies a viscous fourth-order parabolic equation with boundary degeneracy on the basis of Chapter 2.We still apply the regularization method and semi-discrete technique.To deal with viscous term,we use the energy method to obtain better estimates and then use a small parameter limit process to show the existence and uniqueness for weak solutions.Chapter 4 mainly studies the problem of eigenvalues for a fourth-order elliptic operators in high-dimensional space.We give the definition of the eigenvalue and eigenfunction in the sense of distribution.The variational method yields the existence of the eigenvalue and eigenfunction.Finally,we show the several properties for the eigenfunctions and eigenvalues.