Some Properties Of Solutions To Several Pseudo-parabolic Systems

This paper mainly studies some properties of the solutions to three classes of non-linear pseudo-parabolic systems,including local existence,global existence and blow-up property of solution.This thesis is divided into five chapters:the first two chapters are introduction and some basic knowledge.In the introduction,the background and current research for the three parabolic problems 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 systems:In the third chapter of this thesis,we consider the Cauchy problem to the quasi-linear pseudo-parabolic system(3.1).In this chapter,firstly,we use contraction mapping theorem to obtain the local existence of solutions.Then we obtain the blow-up exponent according to the test function method,instead of general Kaplan method for heat system.To obtain the global solutions,we apply the Lp-Lq technique to establish some uniform Lq time-decay estimates.Above all,we establish the critical Fujita exponent(pq)c= 1+(2/N)max(p +1,q +1).In the fourth chapter,we discuss the local solution and global solution of system(4.1).Firstly,we give the related properties of the operators B and J(t)and kernel function for pseudo-parabolic system.Then we obtain the local solution according to the contraction mapping principle.By establishing the couple upper and lower mild solution of the Cauchy problem(4.1),we obtain the global existence of solutions.In the fifth chapter,we study the global existence and blow-up property of solutions to the initial-boundary value problem for a class of pseudo-parabolic system(5.1).We applied the method of Banach fixed point theorem to prove the existence of local solutions.Moreover,we proved the comparison principle of system(4.1)under some suitable condition.Finally we prove the global existence and blow-up property of solutions by using upper and lower solutions method.