Carleman Estimates And Null Controllability For Nonlinear Degenerate Convection-Diffusion Equations

This paper studies the null controllability of the following nonlinear degenerate convection-diffusion system ut-(a(x)ux)x+(p(x,t,u))x+q(x,t,u)=hχω,(x,t)∈ QT,u(0,t)=u(1,t)=0,t ∈(0,T),u(x,0)=u0(x),x ∈(0,1),where QT=(0,1)×(0,T),T>0,h is a control function.χω is the characteristic function of ω=(x0,x1)with 0<x0<x1<1,u0(x)∈L2(0,1),a(x)∈C([0,1])∩ C1((0,1])satisfies a(0)=0;a(x)>0,x∈[0,1],;(?)xa’(x)/a(x)=K;(?)xa’(x)/a(x)<2K+2/3,where the constant K ∈(0,1/2),p and q are measurable functions on(0,1)×(0,T)× R that satisfy|p(x,t,u)-p(x,t,v)|≤M|u-v|,|q(x,t,u)-q(x,t,v)|≤M|u-v|,p(x,t,0)=q(x,t,0)=0,where(x,t)∈ QT,u,v∈R,M>0.The problem considered in this paper has a more general diffusion coefficient a(x),the equation degenerates at x=0,and contains nonlinear convection terms and source terms,where the convection term is not controlled by the diffusion term,all these bring difficulties to the study of the problem.In this paper,the original problem is linearized,the Carleman estimate of the dual problem of the linearized problem is established,and then the observability inequality is obtained.Finally,the null controllability of the nonlinear problem is given.