Several Propertiesof Wellpossedness of a Syestem of Parabolic Equations 

Author  LinXueQing 
Tutor  PanJiaQing 
School  Jimei University 
Course  Applied Mathematics 
Keywords  singular diffusion linear approximation parabolic systems inverse problems wellposedness 
CLC  O175.26 
Type  Master's thesis 
Year  2011 
Downloads  23 
Quotes  0 
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 =△v 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 solution to the nonlinear equation wt = （?）(w^{m1}（?）w) in L^{2}, and the explicit error estimate isobtained:（?）where the positive C_{1}^{*} is independent of T, while C_{1}^{**}depends on T;（4） the solution approaches to u（t） in L^{2} 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 k_{1}, k_{2} > 0 are unknown constants, thereexists a unique solution {k_{1}, k_{2}, u（x, t）, v（x, t）}, which satis?es with the above equations andthe additional conditions:α=∫0^{∞}u（x, t_{1}）dx,β=∫_{0} ^{∞}v（x, t_{1}）dx, and k_{1}, k_{2} are continuousdependent on the additional conditions;Conclusion 2 IF g_{1}（t）, g_{2}（t） are unknown functions, there exists a unique solution{g_{1}（t）,g_{2}（t）,u（x,t）,v（x,t）} which satisfies with the above equations and the additional conditions: h_{1}（t） =∫_{0} ^{∞}u（x, t）dx, h_{2}（t） =∫_{0} ^{∞}v（x, t）dx, and g_{1}（t）, g_{2}（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: h_{1}（t） =∫_{0}^{∞}u（x, t）dx,h_{2}（t） =∫_{0} ^{∞}v（x, t）dx.