Dissertation > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations > Parabolic equation

Several Propertiesof Well-possedness of a Syestem of Parabolic Equations

Author LinXueQing
Tutor PanJiaQing
School Jimei University
Course Applied Mathematics
Keywords singular diffusion linear approximation parabolic systems inverse prob-lems well-posedness
CLC O175.26
Type Master's thesis
Year 2011
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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 solu-tion to the nonlinear equation wt = (?)(wm-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=∫0u(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) =∫0u(x, t)dx,h2(t) =∫0 v(x, t)dx.

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