Vtyjkyuk

I am trying to solve the following problem:

Prove the existence of a solution of the heat equation
$$
u_t=-Delta u +f(t), u(0)=u_0,
$$
with Dirichlet Boundary conditions. Identify the spaces that the data and solution should occupy.

This question is a specific version of question 12.9 in this book http://uxmym1.iimas.unam.mx/ramon/docs/RenRog.pdf

I am looking to use semigroup methods. In particular I am trying to verify the conditions of Lumer-Phillips Theorem, Theorem 12.22 in the book.

Let my operator $A=Delta$. Then if I can show that $A$ satisfies the conditions of Lumer-Phillips I will have that $A$ is the infinitesimal generator of a $C_0$-semigroup and so a solution will exist.

My Hilbert Space is $L^2(Omega)$, and I have that $D(A)=H^2(Omega)cap H^1_0(Omega)$, which is dense in $L^2(Omega)$. So the first condition of Lumer-Phillips is satisfied.

I still need to show the following:
$$
(2) text Re(x,Ax)leq w (x,x), text for some w, text for every xin D(A),
$$
$$
(3) text There exists a lambda_0>w text such that A-lambda_0 I text is onto.
$$
Here $(bullet,bullet)$ denotes the $L^2(Omega)$ inner product, i.e. $(u,v)=int_Omega u(x)overlinev(x) dx$.

I'm not sure how to show (2) or (3) and could use some help.

Thanks!

For $uin D(A)$, an integration by parts yields

beginalign(u,Au)&=int_Omega uoverlineAu;dx
=int_Omega uoverlineDelta u;dx=int_Omega uDeltaoverlineu;dx\
&=sum_i int_Omega uoverlineu_x_ix_i;dx
=-sum_i int_Omega u_x_ioverlineu_x_i;dx
=-sum_i int_Omega |u_x_i|^2;dx\
&=-int_Omega |Du|^2; dxleq 0
endalign

and thus we have $(2)$ with $w=0$.

Let $fin L^2(Omega)$. The sesquilinear form $B:H^1_0(Omega)times H^1_0(Omega)to mathbbC$ given by
$$B[u,v]=int_Omega uoverlinev;dx+int_Omega Ducdot Doverlinev;dx$$
is continuous and coercive. Furthermore, the functional linear $Lambda:H_0^1(Omega)to mathbbC$ given by
$$Lambda(u)=-int_Omega foverlineu;dx$$
is continuous. Thus, by the Lax-Milgram Theorem (see Theorem 1, p. 376, in Dautray's book), there exists an unique $uin H_0^1(Omega)$ such that
$$int_Omega uoverlinev;dx+int_Omega Ducdot Doverlinev;dx=-int_Omega foverlineu;dx,qquadforall vin H_0^1(Omega).$$
It follows from elliptic regularity (see Theorem 9.25 in Brezis book) that $uin H^2(Omega)$ and thus
$$u-Delta u=-f.$$

This argumment shows that, for any $fin L^2(Omega)$, there exists (an unique) $uin D(A)$ such that $Au-Iu=f$. Therefore, $A-I$ is onto and we have $(3)$ with $lambda_0=1$.