The Existence And Uniqueness Of The Generalized Solution For A Class Of Nonlinear Coupled Partial Differential Equations

In this paper,we present the existence and uniqueness of the generalized solution of the following nonlinear coupled system of partial differential equations by basing on the Sobolev space and using the method of Feado-Galerkin u+a1u(4)+bu-(β+M(||u(1)||2+||v(1)||2))u(2)-cu(2)=f(x,t),(x,t)∈(0,l)×(0,T) v+a2v(4)+bu-(β+M(||v(1)||2+||u(1)||2))v(2)-cu(2)=g(x,t),(x,t)∈(0,l)×(0,T) And study the problem of finding u and v solutions,verifying the initial conditions u(x,0)=u0(x),u(x,0)=u1(x),x∈(0,l) v(x,0)=V0(z),v(x,0)=v1(x),x∈(0,l) And the boundary conditions u(0,t)=u(l,t)=u(2)(0,t)=u(2)(l,t)=0v(0,t)=v(l,t)=v(2)(0,t)=v(2)(l,t)=0Here a1,a2,b,c are all numbers of normal,M∈C1([0,∞))is a nonnegative real function about t,β is real, f and g∈L∞(0,T；L2(Ω)),in addition, f and g∈L∞(0,T；L2(Ω)).We obtained the existence and uniqueness of weak solution and strong solution of the equations under the certain initial and boundary conditions.In the following conditions:u0,v0∈V,u1,v1∈L2(Q).f,g∈L∞(0,T；L2(Ω)). Function M meets the conditions:M’(λ)≥0and m0,m1>0,so that M(λ)≥m0+m1λ,we proved the existence and uniqueness of weak solution of the system. In the following conditions:u0,v0∈V1and u1,v1∈V. And f,g,f and g∈L∞(0,T;L2(Ω)). Function M meets M’(λ)≥0and m0,m1>0, so that M(λ)≥m0+m1λ, we proved the existence and uniqueness of strong solution of the system.