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: $$ (2) \text{ Re}(x,Ax)\leq w (x,x), \text{ for some } w, \text{ for every }x\in D(A), $$ $$ (3) \text{ There exists a } \lambda_0>w \text{ such that } A-\lambda_0 I \text{ is onto.} $$ Here $(\bullet,\bullet)$ denotes the $L^2(\Omega)$ inner product, i.e. $(u,v)=\int_{\Omega} u(x)\overline{v(x)} dx$.
I'm not sure how to show (2) or (3) and could use some help.
Thanks!
For $u\in D(A)$, an integration by parts yields
\begin{align}(u,Au)&=\int_\Omega u\overline{Au}\;dx =\int_\Omega u\overline{\Delta u}\;dx=\int_\Omega u\Delta\overline{u}\;dx\\ &=\sum_{i} \int_\Omega u\overline{u}_{x_ix_i}\;dx =-\sum_{i} \int_\Omega u_{x_i}\overline{u}_{x_i}\;dx =-\sum_{i} \int_\Omega |u_{x_i}|^2\;dx\\ &=-\int_\Omega |Du|^2\; dx\leq 0 \end{align}
and thus we have $(2)$ with $w=0$.
Let $f\in L^2(\Omega)$. The sesquilinear form $B:H^1_0(\Omega)\times H^1_0(\Omega)\to \mathbb{C}$ given by $$B[u,v]=\int_\Omega u\overline{v}\;dx+\int_\Omega Du\cdot D\overline{v}\;dx$$ is continuous and coercive. Furthermore, the functional linear $\Lambda:H_0^1(\Omega)\to \mathbb{C}$ given by $$\Lambda(u)=-\int_\Omega f\overline{u}\;dx$$ is continuous. Thus, by the Lax-Milgram Theorem (see Theorem 1, p. 376, in Dautray's book), there exists an unique $u\in H_0^1(\Omega)$ such that $$\int_\Omega u\overline{v}\;dx+\int_\Omega Du\cdot D\overline{v}\;dx=-\int_\Omega f\overline{u}\;dx,\qquad\forall\ v\in H_0^1(\Omega).$$ It follows from elliptic regularity (see Theorem 9.25 in Brezis book) that $u\in H^2(\Omega)$ and thus $$u-\Delta u=-f.$$
This argumment shows that, for any $f\in L^2(\Omega)$, there exists (an unique) $u\in D(A)$ such that $Au-Iu=f$. Therefore, $A-I$ is onto and we have $(3)$ with $\lambda_0=1$.