Existence and uniqueness for evolutionary problem.

roposition

(Existence and uniqueness for evolutionary problem) Suppose that the bilinear form is defined by ( Bilinear form B 2 ), the condition ( Symmetric principal part ) holds and MATH MATH MATH MATH Then there exists a unique solution of the problem ( Evolutionary variational inequality problem ) and MATH

Proof

(Uniqueness) Suppose existence of two solutions $u_{1}$ and $u_{2}$ of the problem ( Evolutionary variational inequality problem ): MATH

We set $u=u_{1}$ and $v=u_{2}$ and then we also set $u=u_{2}$ and $v=u_{1}$ : MATH and add: MATH Let $w=u_{1}-u_{2}$ then we have MATH According to the proposition ( Energy estimates for the bilinear form B )-1: MATH thus MATH We integrate over $\left[ t,T\right]$ and use to deduce MATH We drop the positive term : MATH Assume that . Let $t^{\ast}$ be the rightmost local maximum of on $\left[ 0,T\right]$ so that is decreasing on and . Then application of the mean value theorem leads to MATH which leads to contradiction with the decreasing behavior of as we put $h\downarrow0$ . Therefore .