The Qualitative Analysis For Several Differential (Integral) Systems

It is well known that nonlinear differential(integral)equations play a crucial role in differential geometry,applied science and fluid mechanics etc.Although it is very difficult to solve the nonlinear system completely,many scholars have found many effective methods for some special nonlinear systems.In this thesis,we mainly study several qualitative properties of solutions for some nonlinear differential(integral)systems,which can be mainly divided into two parts:the first part is the existence,stability and decay estimate of the solutions for a class of elliptic problems;the sec-ond part is the existence and stability of peaked solutions for some shallow water equations.The details are as follows:In Chapter 1,we review some notations and conventions,and give several pre-liminary results which will be used in later chapters.In the first part of the Chapter 2,we study the stability and decay estimate of the regular solutions for k-Hessian equation with radial structurewhere n? 3,1<k<n/2,p>nk/n-2k.And in the second part,we prove the boundary blowing-up estimate of the general k-Hessian equation?k(D2V)=(-V)p,x??,where ? is a bounded doamin in Rn and p>1,n? 3,1<k<n/2.Here?k(D2V)=Sk(?(D2V)),?(D2V)=(?1,?2,…,?n),with ?i being eigenvalues of the Hessian matrix(D2V)and Sk(·)is the k-th symmetric functionUsing "Doubling lemma",we extend the results in cases of semi-linear and quasi-linear to fully nonlinear elliptic systems.In Chapter 3,we study the radial symmetry,monotonicity and regularity lifting of positive solutions for the following Wolff type integral equation u(x)=c(x)W?,?(up)(x),u>0,v?Rn,where c(x)is a double bounded(namely,there exist positive constants c and C such that c ?c(x)?C for all x ?Rn),andFirstly,using the "Doubling lemma",we prove the regularity lifting result for this equation when ?>1,p>?-1,0<??<n.That is to say,if(?),is a positive solution of this equation,then u ? L?(Rn).Based on this result,we also obtain the nonexistence of integrable solution and finite energy solution for k-Hessian equation when nk/n-2k<p<n(p-k)/n-2k.Secondly,using the method of moving planes in integral form,we prove that the positive solutions of this equation are radially symmetric and decreasing about some point in Rn when c(x)?1,??(1,2],p>n(?-1)/n-??,0<??<n.In Chapter 4,we investigate the existence and stability of periodic peakons for the following initial value problem to Novikov equation with cubic nonlinearity Here S=[0,1).By the method of defining weak solutions via test functions,we prove that the periodic peakons are global weak solutions.In Chapter 5,we are concerned with the existence of peaked solutions for the following Novikov-CH equation with quadratic and cubic nonlinearities mt+k1(3uuxm+u2mx)+k2(2mux+mxu)=0,m=u-uxx,t>0.Using related Green's functions and the method of solving weak solutions of nonlin-ear partial differential equations,we deduce the single peaked solutions and periodic peaked solutions.Some simulations also have been given.