In this paper, a positive definite semi-discrete mixed finite element method was presented for two-dimensional parabolic equations. In the new positive definite systems, the gradient equation and flux equations were separated from their scalar unknown equations.  Also, the existence and uniqueness of the semi-discrete mixed finite element solutions were proven. Error estimates were also obtaine...

AMS Mathematics Subject Classification (2000): 47D06, 47D60, 47D62

This paper is concerned with the numerical solution of parabolic equations coupled to gradient equations. The gradient equations are ordinary diierential equations whose solutions deene positions of particles in the spatial domain of the parabolic equations. The vector eld of the gradient equations is determined by gradients of solutions to the parabolic equations. Such mixed parabolic-gradient...

PARABOLIC SYSTEMS 185 such that u = F(F 1u). We de ne u(A) 2 B(X) by (1:3) u(A)x = ZRn(F 1u)( )e i( ;A)x d for x 2 X: De ne MN (FL1) = f(ujk); ujk 2 FL1g. Similarly, MN(Lp) and so on. If u = (ujk) 2 MN(FL1) then u(A) (ujk(A)) 2 B(XN ). It is known that MN (FL1) is a (non-commutative) Banach algebra under matrix pointwise multiplication and addition with norm kukFL1 kF 1ukL1 , where F 1u = (F 1u...

‎This note deals with the systems of parabolic equations with local and localized sources involving $n$ components‎. ‎We obtained the exponent regions‎, ‎where $kin {1,2,cdots,n}$ components may blow up simultaneously while the other $(n-k)$ ones still remain bounded under suitable initial data‎. ‎It is proved that different initial data can lead to different blow-up phenomena even in the same ...

We consider a class of higher order nonlinear degenerate parabolic systems, whose easiest model is the parabolic p-Laplacean system ∫ ΩT ( u · φt − |Dmu|p−2Dmu ·Dmφ ) dz = ∫