Oscillation About Non-trivial Solution Of Partial Functional Differential Equations

In this paper, oscillation of solution about non-trivial solution for three classes of partial functional differential equations are studied. In chapter two, oscillation of solution about non-trivial steady state for a nonlinear delay parabolic equations are discussed. By the first order delay differential inequalities and eigenvalue problem, several sufficient criteria of oscillation of the equation about the non-trivial equilibrium are obtained under three different boundary conditions. One example is given to show the applicability of the proposed approach. In chapter three, the oscillation of solutions about non-trivial steady state for the systems of a nonlinear delay parabolic equations are investigated. By using averaging method, vertical adding, sgn function, Green's formula, Lipschits condition and Lagrange mean value theorem, the oscillatory problems for the systems of parabolic equations are reduced to the problem that the differential inequality has eventually positive solution or not. Moreover, some sufficient conditions for oscillation about non-trivial steady state of all solutions are respectively obtained under three boundary value conditions. In chapter four, oscillation of solution about non-trivial steady state for a nonlinear delay hyperbolic equations are studied. Removing the reaction-diffusion by using Green's formula, and turning the second-order functional differential inequalities into first-order functional differential inequalities, some sufficient conditions for oscillation about non-trivial steady state of all solutions are respectively obtained under three boundary value conditions.