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I am working on a problem from a textbook and have run into difficulties on this specific question. Any assistance will be appreciated,

Consider the partial differential equation,

$\frac{\partial z}{\partial t}(x,t) = \frac{\partial^2z}{\partial x^{2}}(x,t) - \alpha\frac{\partial z}{\partial x}(x,t)$

$\frac{\partial z}{\partial x}(0,t) = \frac{\partial z}{\partial x}(1,t) = 0$

$z(x,0) = z_{0}(x)$

a) Formulate this as an abstract system on the state space $L_{2}(0,1)$ with both the usual inner product and the weighted inner product

$\langle z_{1} , z_{2} \rangle_{a} = \int_{0}^{1} z_{1}(x)\overline{z_{2}(x)}\exp(-\alpha x)dx $

For this part, I have this solution (I have confirmed this is correct)

$Ah = \frac{d^{2}h}{dx^{2}} - \alpha\frac{dh}{dx}$

$D(A) = \{ h \in L_{2}(0,1) | h , \frac{dh}{dx} \text{ are absolutely continuous,}$

$\frac{d^{2}h}{dx^{2}} \in L_{2}(0,1) \text{ and } \frac{dh}{dx}(0,t) = \frac{dh}{dx}(1,t) = 0 \}$

b) Show that $A$ generates a contraction semigroup $T(t)$ on $L_{2}(0,1)$ with the weighted inner product.

I am stuck as to how to approach the second part of this question. I will appreciate any guidance or an appropriate source for guidance,

Thank you for your time

This problem appears in:

An Introduction to Infinite-Dimensional Linear Systems Theory Ruth F. Curtain, Hans Zwart

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1 Answer 1

up vote 0 down vote accepted

The details will depend on what version of the Lumer-Phillips theorem your textbook presents. But you will certainly need to show that $A$ is dissipative: that is, $\langle Au,u\rangle\le 0$ for all $u\in D(A)$. This should not be hard: split the integral $$\int_0^1 (h''(x)h(x)\,e^{-\alpha x} - \alpha h'(x)h(x)\,e^{-\alpha x} )\,dx$$ in two, integrate the first one by parts and simplify.

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