Qualitative Analysis Of Nonlinear Pseudo-parabolic Equation

The aim of this paper is to investigate two classes of nonlinear pseudo-parabolic partial equations and obtain their solutions.To solve the nonlinear pseudo-parabolic partial equation which left side is quasi-linear, this paper combines Riemann's method with the fixed point theory and obtains the existence and uniqueness of the classical solution to this problem. First the left side of the equation is turned into a linear operator. Then by using the Riemann's function of the operator and constructing an auxiliary problem, an integrodifferential equation equivalent to former problem is got. At the end, through a mapping and Banach fixed point theory, the existence and uniqueness of the classical solution to this integrodifferential equation is obtained. That is, there is one and only one classical solution to the nonlinear pseudo-parabolic partial equation.As for mixed problem with integral boundary conditions for a certain nonlinear pseudo-parabolic partial equation, the existence and uniqueness of a strong solution to the corresponding linear problem is established and so the nonlinear problem can be solved by using addition theorem. And to solve the linear problem is turned to obtain the solution of an operator equation equivalent to it. Firstly, this paper introduces a function space defined by Bouziani and takes the inner product in space L2 of the linear equation and an integrodifferential operator. Secondly, through integrating by parts and using Cauchy inequality, thanks to a lemma of Gronwall's type, an energy inequality is got and a priori estimate is established. So there is one and only one solution to the corresponding linear pseudo-parabolic partial equation can be proved by using the priori estimate and the density of the range of the operator. Finally by constructing an auxiliary problem, using addition theorem and applying an iterative process based on results obtained for the linear problem, the existence and uniqueness of the weak generalized solution of the nonlinear problem is proved.