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Section 9.9 Application to PDEs

In this section we study a number of partial differential equations that are relevant in many other areas of applied mathematics, such as reaction-diffusion equations, wave equations, the Schr\"odinger equation, and integral equations. We show how the semigroup theory can be used to find solutions for these models.

Subsection 9.9.1 The Reaction Diffusion Equation

On a smooth domain \(\Omega\subset \RR^n\) we consider a reaction-diffusion equation
\begin{equation} \begin{array}{rcll} u_t \amp = \amp \Delta u + f(t) \amp \quad \mbox{on} \quad\Omega, \\ u(x,0) \amp=\amp u_0 \amp \quad \in L^2(\Omega), \\ u(x,t) \amp=\amp 0 \amp \quad x\in \partial \Omega, \end{array}\label{RDE}\tag{9.9.1} \end{equation}
with \(f\in C^0([0, \infty])\text{.}\) We chose the Hilbert space \(X=L^2(\Omega)\) and the domain for \(A=\Delta\) as
\begin{equation*} D(A) = H^2(\Omega)\cap H_0^1(\Omega). \end{equation*}
This set is dense in \(L^2(\Omega)\text{.}\) On this Hilbert space \(L^2(\Omega)\) the operator \(\Delta \) is self-adjoint and for \(u\in D(A)\) we have
\begin{equation*} (u, \Delta u) =\int_\Omega u \Delta u dx = -\int_\Omega\nabla u \nabla u dx = \int \Delta u u ds = (\Delta u, u) . \end{equation*}
This also shows that
\begin{equation*} \mbox{Re}(u, \Delta u) \leq 0 . \end{equation*}
Hence we chose the spectral bound in the Lumer-Phillips Theorem as \(\omega =0\text{.}\) Now we choose a \(\lambda_0>0\) and show that \((A-\lambda_0 I)\) is onto. Given \(\varphi\in L^2(\Omega)\) we aim to find \(u\in D(A)\) such that
\begin{equation*} \Delta u - \lambda_0 u = \varphi, \qquad u|_{\partial\Omega} =0 . \end{equation*}
This leads to the theory of elliptic partial differential equations on a smooth domain with Dirichlet boundary conditions, which are solved elsewhere (see Gilbarg and Trudinger \cite{GT}). Here we simply assume that such an \(\lambda_0\) exists. Then we can apply Lumer-Phillips to \((A, D(A))\text{,}\) and \(A\) generates a strongly continuous contraction semigroup \(e^{\Delta t }\text{.}\) A solution of the reaction-diffusion equation ((9.2.1)) can then be found using the variation of constant formula:
\begin{equation*} u(t) = e^{\Delta t} u_0 +\int_0^t e^{\Delta (t-s)} f(s) ds \end{equation*}
and, using the same arguments as we did in the discussion of mild solutions, we have
\begin{equation*} u\in C^1([0,\infty), L^2(\Omega))\cap C^0([0,\infty), D(A)). \end{equation*}

Subsection 9.9.2 The Wave Equation

On a smooth bounded domain \(\Omega\subset\RR^n\) we consider the wave equation
\begin{equation} \begin{array}{rcll} u_{tt}\amp=\amp \Delta u \amp \quad \mbox{on} \quad\Omega, \\ u(x,t) \amp=\amp 0 \amp\quad \mbox{on } \partial \Omega,\\ u(x,0) \amp=\amp u_0(x), \amp \\ u_t(x,0) \amp=\amp u_1(x). \amp \end{array}\label{waveE}\tag{9.9.2} \end{equation}
We introduce a variable for the velocity \(v(x,t) = u_t(x,t)\) and write the wave equation as a system for \((u,v)\) as
\begin{equation} \left(\begin{array}{c} u \\ v \end{array}\right)_t = \left(\begin{array}{cc} 0 \amp I \\ \Delta \amp 0 \end{array} \right) \left(\begin{array}{c} u \\ v \end{array}\right).\tag{9.9.3} \end{equation}
This is a differential equation on \(X= H_0^1(\Omega) \times L^2( \Omega)\text{.}\) The matrix will become our generator with the following domain:
\begin{equation*} A := \left(\begin{array}{cc} 0 \amp I \\ \Delta \amp 0 \end{array} \right), \qquad D(A) =\Bigl(H^2(\Omega)\cap H_0^1(\Omega) \Bigr)\times H^1(\Omega), \end{equation*}
and the domain is dense.
To apply the Lumer-Philips result, we consider the inner product on \(X\)
\begin{equation*} \langle \left(\begin{array}{c} u\\v\end{array} \right), \left(\begin{array}{c} f\\g\end{array} \right)\rangle = \int_\Omega \nabla u \nabla f + v g \; dx . \end{equation*}
Then
\begin{equation} \langle \left(\begin{array}{c} u\\v\end{array} \right), A \left(\begin{array}{c} u\\v\end{array} \right) \rangle = \langle \left(\begin{array}{c} u\\v\end{array} \right), \left(\begin{array}{c} v\\ \Delta u\end{array} \right)\rangle = \int\nabla u \nabla v + v \Delta u \; dx = 0 .\tag{9.9.4} \end{equation}
Hence we chose the spectral bound as \(\omega =0\text{.}\) Next we claim that each \(\lambda>0\) satisfies \(\lambda\in \rho(A)\text{.}\) Given \((f,g)\in H_0^1(\Omega)\times L^2(\Omega)\) we solve for \((u,v)\) the equation
\begin{equation} (A-\lambda I) \left(\begin{array}{c} u\\v\end{array} \right) = \left(\begin{array}{c} f\\g\end{array} \right). \tag{9.9.5} \end{equation}
We get two equations
\begin{equation} v - \lambda u = f, \qquad \Delta u - \lambda v = g. \tag{9.9.6} \end{equation}
Solving the first equation for \(v\) and substituting this into the second equation gives
\begin{equation*} \Delta u -\lambda^2 u = g +\lambda f, \end{equation*}
where \(g+\lambda f\in L^2(\Omega)\text{.}\) Again we employ elliptic solution theory \cite{GT} and get a unique solution for \(\lambda^2>0\text{.}\) This allows us to apply the Lumer-Phillips Theorem and obtain a solution semigroup of the wave equation
\begin{equation*} T(t) = \exp\left(\left(\begin{array}{cc} 0 \amp I \\ \Delta \amp 0 \end{array} \right)t\right). \end{equation*}

Subsection 9.9.3 The Schr\"odinger Equation

The Schr\"odinger equation for a given real potential \(V(x)\) in \(\RR^n\) is
\begin{equation*} u_t = i(\Delta u - V(x) u), \end{equation*}
where \(u(x,t)\) is complex valued wave function.
We compute
\begin{align*} (u, Au ) \amp=\amp \int u\; \overline{i(\Delta u - V(x) u ) } dx\\ \amp=\amp \int u (-i(\Delta \bar u - V(x) \bar u)) dx\\ \amp=\amp \int -i u \Delta \bar u + i V(x) u \bar u dx\\ \amp=\amp \int -i (\Delta u - V(x) u) \bar u dx\\ \amp=\amp -(Au,u). \end{align*}
If \(V(x)\) is bounded, then \(A\) is closed and Stones Theorem for skew-adjoint operators applies \cite{pazy} and we get a contractions semigroup. If \(V(x)\) is unbounded, then all hell breaks loose and we quickly enter areas of active research.

Subsection 9.9.4 Integral Equations

On a bounded domain \(\Omega\subset \RR^n\) we consider the integral equation
\begin{equation*} u_t =\int_\Omega k(x,y) u(y,t) dy =Au, \end{equation*}
with a kernel \(k(x,y)\in L^2(\Omega\times\Omega)\) and \(\int_\Omega k(x,y) dx =1\text{,}\) and with domain of definition
\begin{equation*} D(A) = L^2(\Omega)=X. \end{equation*}
Then \(A\) is a closed, compact Hilbert-Schmidt operator with full domain. From spectral theory we know that \(A\) has a discrete point spectrum and no other parts of the spectrum. This means that the resolvent set is dense in \(\CC\text{,}\) and we happily find one \(\lambda_0\) such that \(A-\lambda_0 I\) is onto. We estimate the norm
\begin{align*} |(u, Au)| \amp=\amp\Bigl| \int_\Omega\int_\Omega k(x,y) u(x) u(y) dx dy \Bigr|\\ \amp\leq \amp \Bigl| \int_\Omega \|k(\cdot, y)\|_2 \|u\|_2 u(y) dy \Bigr|\\ \amp\leq \amp \|k\|_2 \|u\|_2\|u\|_2\\ \amp=\amp \|k\|_2 \|u\|_2^2. \end{align*}
Hence we choose \(\omega = \|k\|_2 \text{.}\) By Lumer-Phillips, \(A\) generates a quasi contraction semigroup
\begin{equation*} \|T(t) \|\leq e^{\omega t}. \end{equation*}