Some Problems Of Two Camassa-Holm Type Equations

Wave equation is a kind of development equation,which describes the process of system changing with time in many natural sciences.A large number of nonlinear mathematical problems have been proposed in many scientific fields such as physics,biology and ecology.These problems are often studied by establishing nonlinear parabolic equations and nonlinear wave equations in mathematical models.Different nonlinear terms in the equation may lead to the singularity of the solution,such as the blow-up,quenching,extinction,wave-breaking and other phenomena of the solution in a finite time.A wave breaking will arise as soon as the initial value decay faster than the solitary waves at infinity.Therefore,the wave-breaking phenomena is studied by three different methods we provided of the solution for the weakly dissipative.In addition,we exhibit the persistence results of the solution in weighted LP-spaces.Finally,we provide a blow-up criterion for CH-mCH-NOvikov equation.The main research contents of the thesis are as follows:(?)the wave-breaking phenomena of the weakly dissipative shallow water equation is studied by three different methods we provided,which extends the information of this equation.Since the presence of the weakly dissipative term,the equation loses the conservation law E=?Ru2+ux2dx.This difficulty has been solved by establishing the energy inequality.The main idea of the first two methods is to construct the Riccati-type differential inequality to show that the solution of the equation blows up in finite time.The last method depends on the introduction of two families of Lyapunov functions.(?)we exhibit the persistence results of the solution of the weakly dissipative shallow water equation in weighted Lp-spaces.We first introduce some preliminary knowledge to study persistence,and then prove that the solution of the equation decays continuously when the initial value decays continuously.(?)we study the blow-up criterion of CH-mCH-Novikov equation.The main research method is to use Gronwall inequality,1-D Moser-type estimation and transport equation theory in Hs space to obtain the blow-up criterion of the equation by induction of exponential s.