Well-possedness Of Inverse Problem For A Nonlinear Parabolic Equation

There are three parts in this thesis. The first one is the existence of singular diffusion equation. The second one is the existence, uniqueness and stability of the solution for inverse problem of degenerate parabolic equation. The third one is the numerical solutions for inverse problem of parabolic equation.1.Consider the second boundary value problem for singular diffusion equation: ut=△um-cup(0<m<1,p>0,c>0) Conclusion:The local existence of the solution, and the sufficient and necessary condition for the existence of global solution.2. Consider the inverse problem with unknown coefficient C for degenerate parabolic equation: ut=△um-cu(m>1,c>0) Conclusion:If c is a constant, under the additional condition∫Ωu(x,T)dx=α, we obtain the explicit expression of C and the existence, uniqueness and stability of the solution; If c is a function, under the additional condition∫Ωu(x,T)dx=h(t), we prove the existence of the solution.3. Consider the numerical solutions for inverse problem of parabolic equation with unknown coefficient c(t) ut=(um)xx+c(t)u{m≥1) under the additional condition u(l/2,t)=g(t).Conclusion:The existence, stability and convergence of the solution are proved, and the order of convergence is given.