# Uniqueness for ode coming from parabolic pde

1. Apr 20, 2009

### cliowa

Hey all,
I was working a little on parabolic pde, and came across this (comes up in regularity theory). Consider a Hilbert triple $V\subset H\subset V^*$ (continuous embeddings) and a linear operator $A(t)$ from V to V*, where t ranges in some interval [0,T]. Now let $w\in H^1(0,T;V^*)\cap L^2(0,T;V)$ solve

$$w'=A(t)w-\int_0^t A'(\tau)w(\tau) d\tau, \quad w(0)=0$$.

I want to show that this implies w=0. How could I do that?

I tried multiplying by w and integrating by parts, which results in

$$1/2 (w,w) +\int_0^t a(s,w(s),w(s))ds= -\int_0^t (\int_0^s A(\tau)w(\tau d\tau,w(s))ds,$$

where a(s,w(s),w(s)) is the induced quadratic form satisfying $a(s,w(s),w(s))\geq \alpha ¦w(s)¦_V-\beta ¦w(s)¦_H$ for constants >0, uniformly in t. How does this help me?

Best regards...cliowa