Some Properties Of Solutions To Several Pseudo-parabolic Equations And Pseudo-parabolic Systems

This thesis mainly studies some properties of the solutions to three classes of pseudo-parabolic equations and system,including global existence,blow-up,initial energy and so on.This thesis is divided into five chapters,in the first two chapters,we present introduction and some basic knowledge.In the introduction,the background and current research for the three pseudo-parabolic equations are given.In the second chapter,some basic knowledge to be used in this thesis is given.In the third,fourth and fifth chapter,we separately consider the following systemsIn the third chapter,we consider the blowup property of solutions to the pseudo-parabolic system with localized nonlinear terms(3-1).First of all,by using constructing mapping and the fixed point theory,we obtain the existence and uniqueness of local solutions.The argument we use to prove the blow up property is based on the comparison principle and probability density of kernel function H.The key point is to construct a pair of suitable comparison functions.For the problem(3-1),this argument allows us to eliminate the assumption of convexity on f1(s),f2(s),which is important in proving the finite time blow up for the usual nonlinear heat equations.In the forth chapter,by the use of the potential well method,we discuss a class of pseudo-parabolic equation with the p-Laplace operator(4-1).We first introduce a family of potential wells and its corresponding sets.Then we define the weak solution.When J(u0)= d and I(u0)?0,we prove the global existence of weak solutions.Moreover,we obtain finite time blow-up when J(u0)= d and I(u0)<0.In the fifth chapter,we consider the blow-up properties of solutions to pseudo-parabolic system with nonlinear nonlocal sources(5.1).We consider the integral of the nonlocal in time.First of all,by using constructing mapping and the fixed point theory,we obtain the existence and uniqueness of local solutions.Then we present the maximum principle and comparison principle of the Cauchy problem.Secondly,we obtain that the solutions blow up relys on a variant of the eigenfunction method combined with new properties on systems of differential inequalities.Finally,we present the special case of the model and the corresponding proof of it.