Several Propertiesof Well-possedness Of A Syestem Of Parabolic Equations

This thesis includes two parts. The ?rst one is the Neumann boundary value problemof singular di?usion equation with convection. The second one is a inverse problems for thelinear parabolic systems with unknown coe?cients, unknown boundary values and unknownsources.I. Consider the problem:(?)We proved that:(1) there exits a unique solution to the problem ;(2) the solution u(x, t, m, p) converges to its corresponding solution u(x, t, 1, 0),whichis the solution to the linear equation vt =â–³v in L2 as m→1, p→0, and the expliciterror estimate is obtained:(?)(3) the solution u(x, t, m1, p) converges to the solution w(x, t, m2), which is the solu-tion to the nonlinear equation wt = (?)(w^m-1(?)w) in L², and the explicit error estimate isobtained:(?)where the positive C₁^* is independent of T, while C₁^**depends on T;(4) the solution approaches to u(t) in L² as t→∞, and the explicit estimate is given:(?)II. Consider the parabolic system which is composed of(?) Conclusion 1 For the given t1∈(0,T), If k₁, k₂ > 0 are unknown constants, thereexists a unique solution {k₁, k₂, u(x, t), v(x, t)}, which satis?es with the above equations andthe additional conditions:α=∫0^∞u(x, t₁)dx,β=∫₀ ^∞v(x, t₁)dx, and k₁, k₂ are continuousdependent on the additional conditions;Conclusion 2 IF g₁(t), g₂(t) are unknown functions, there exists a unique solution{g₁(t),g₂(t),u(x,t),v(x,t)} which satisfies with the above equations and the additional condi-tions: h₁(t) =∫₀ ^∞u(x, t)dx, h₂(t) =∫₀ ^∞v(x, t)dx, and g₁(t), g₂(t) are continuous dependenton the additional conditions;Conclusion 3 Consider the parabolic system which is composed of(?)Ifφ1(t),φ2(t) are unknown functions, then there exists a unique solution {φ1(t),φ2(t), u, v}which satisfies with the above equations and the additional conditions: h₁(t) =∫₀^∞u(x, t)dx,h₂(t) =∫₀ ^∞v(x, t)dx.