Existence Of Two Nonlinear Fourth-order Parabolic Equations

The nonlinear fourth order parabolic equation is one important topic in partial differential equations.The typical model includes the population model,Cahn-Hilliard equation and thin film equation,which have a lot of applications in the population problem,phase transition theory,film lubrication theory,fluid theory,chemical and economic problems etc.The paper mainly studies the existence of two nonlinear fourth order parabolic equations.The main content is as following.The first part,the existence of weak solutions of a viscous fourth order degenerate parabolic equation is studied(?)u/(?)t-k(?)?u/(?)t-+?(|?u_p-2?u)= f(x,t),x ? ?,t>0 u = ?u = 0,x ?(?)Q,t>0,u(x,0)= u0(x),x??,where ?(?)RN is a bounded domain with smooth boundary.p>2 is a constant.u0(x)is the initial function.k>0 is the viscosity coefficient and k(?)?u/(?)t denotes the viscosity term.?(|?u_p-2?u)=?p2u is called p-biharmonic operator.We consider a corresponding semi-discrete problem at first.The existence of weak solutions of the corresponding elliptic problem is obtained by using minimizer functional method.Secondly,some approximation solutions of the fourth order degenerate parabolic equation are constructed.By choosing reasonable test functions to get the necessary uniform estimations,the convergence results can be obtained.Finally the existence of the weak solutions of the problem is proved.The second part,the initial boundary value problems of a population model is studied in 3 dimensional space ut=-?(a1(x)?u)+ a2?u +a?u3 +G(u),(x,t)?QT u= 0,?u = 0,(x,t)?(?)?×[0,T]u(x,0)= u0(x),x? ?,where Q is bounded domain of R3 with smooth boundary(?)?,QT =?×(0,T),T>0,0<m<a1(x)<M,a>0,a2=0 are constants,u0(x)is known function,G(s)is a nonlinear function.In order to study the existence of the solutions,the Galerkin method is applied to construct the necessary approximate solution.The uniform estimates of the approximate solution give the convergence limits and then the existence of the weak solutions of the corresponding problem is proved.