Global Existence And Blow-up Of Solutions For A Class Of Higher Order Equations

In the thesis,we study some properties of solutions for a class of high order equations,including existence and uniqueness of weak solutions,blow-up,extinction and non-extinction.Firstly,we study the initial-boundary-value problem for st isothermal viscous Cahn-Hilliaxd equation with inertial term,which arises in isothermal fast phase sepa-ration processes.where ? Rn(n ?3)is a bounded domain with smooth boundary,?>0 is a inertialparameter,k? 0 is a viscosity coefficient and f(s)is a given nonlinear function.Based on the Galerkin method and the compactness theorem,we establish the existence of the global generalized solution.To prove the blow-up of solutions,we establish a new functional and consider the solution of the Bernoulli type equation.Basing on the required estimates and using a lemma on the ordinaxy differential inequality of a second order,we prove the blow-up of the solution for the initial-boundary problem.Secondly,we consider the initial-boundary-value problem for a viscous Cahn-Hilliard type equation with inertial term in one spatial dimension,which arises in dynamics of phase transitions in ternary oil-water-surfactant systems.where ?=(0,l)(?)R is a bounded domain.We show that the dynamical system gen-erated by this problem admits a global attractor in the phase space H03(0,l)×L2(0,l).Thirdly,we consider an initial-boundary-value problem for a class of p-biharmonic parabolic equation with logarithmic nonlinearity in a bounded domain.where ? is a bounded domain in Rn with smooth boundary a?,p?q are positive constants,and u0?(W01,p(?)?W2,p(?))\{0}.We obtain three relatively complete conclusions:if 2<p<g<p(1+4/n)and u0 ? W+,the problem has a global weak solutions;if 2<p<q<p(1+4/n)and u0 ?w1-,the solutions blow-up at finite time;and we also obtain the results of blow-up,extinction and non-extinction of the solutions when max{1,2n/n+4}<p?2.Finally,we study the following Cauchy problem of tuhe degenerate convective Cahn-Hilliard equation and we investigate the existence of solutions.To prove the existence of classical solu-tions,the main difficulties are caused by the equation which is degenerate in the x1 direction and the nonlinearity of ?x'2A(u).Our method is based on the long-short wave method and the frequency decomposition method.To estimate the low frequency part,we use Green's function methods,and to deal with the high frequency part,we employ energy estimates and Poincare-like inequality.For the standard continuity argument,we obtain the local solution first and then extend it to a global in time solution by establishing the uniform estimates of the solution.