<!-- NOTATION PROPOSAL: For $C_0$-semigroups, we use $\{T(t)\}_{t \ge 0}$ for the semigroup family, $A$ for the infinitesimal generator, and $D(A)$ for its domain. The notation $T(t) \in \mathcal{L}(X)$ follows the existing bounded-operator standard. Analytic semigroups in a sector $\Sigma_\delta = \{z \in \mathbb{C} \setminus \{0\} : |\arg z| < \delta\}$ need the sector notation which is new to the notation guide. --> ## Semigroup Theory Consider the heat equation on a bounded domain $U \subset \mathbb{R}^n$ with Dirichlet boundary conditions: \begin{align*} \partial_t u - \Delta u &= 0 \quad \text{in } U \times (0, \infty) \\ u &= 0 \quad \text{on } \partial U \times (0, \infty) \\ u(\cdot, 0) &= g \quad \text{in } U. \end{align*} A classical student's first instinct is to seek an explicit formula — and indeed, for the heat equation on all of $\mathbb{R}^n$, one can write $u(x, t) = (K_t * g)(x)$ where $K_t$ is the heat kernel. But what happens when the domain is curved, the boundary conditions are complex, or the operator is not the Laplacian but some other second-order elliptic operator $L$? The explicit formula breaks down, and we are left with a question: is there a framework that treats the evolution $g \mapsto u(\cdot, t)$ as a coherent family of operators, parameterized by time, without requiring an explicit kernel? This is precisely what the theory of $C_0$-semigroups provides. Instead of asking for $u(x, t)$ directly, we ask: what is the map $t \mapsto u(\cdot, t)$ as a curve in a Banach space $X$? If we write $u(t) := u(\cdot, t) \in X$ for a suitable function space $X$ (say $X = L^2(U)$ or $X = C(\bar{U})$), then the heat equation becomes the abstract ordinary differential equation \begin{align*} \frac{d}{dt} u(t) &= Au(t), \quad t > 0 \\ u(0) &= g, \end{align*} where $A = \Delta$ acts as a (unbounded) operator on $X$. The "solution" should then be $u(t) = e^{tA} g$ in some sense — and the entire theory of $C_0$-semigroups is the rigorous construction of this exponential map for operators that may be unbounded, such as differential operators on function spaces. The payoff is enormous: once we understand the abstract framework, we can treat not just the heat equation but also the wave equation, Schrödinger's equation, linear transport equations, and a host of other evolution equations in a unified way. The generator $A$ of the semigroup encodes the PDE, the domain $D(A)$ encodes the boundary conditions, and the semigroup $\{T(t)\}_{t \ge 0}$ encodes time evolution. The celebrated Hille-Yosida theorem then tells us exactly which operators $A$ can arise as generators — turning the existence theory for evolution equations into a spectral question about $A$. [example: The Heat Semigroup on $L^2$] To fix ideas, consider $X = L^2(U)$ where $U \subset \mathbb{R}^n$ is a bounded open set with smooth boundary, and let $A = \Delta$ with domain $D(A) = H^2(U) \cap H^1_0(U)$. The spectral theorem for the Laplacian guarantees a sequence of eigenfunctions $\{e_k\}_{k=1}^\infty$ (normalized in $L^2(U)$) with eigenvalues $-\lambda_k < 0$ satisfying \begin{align*} -\Delta e_k &= \lambda_k e_k \quad \text{in } U \\ e_k &= 0 \quad \text{on } \partial U. \end{align*} For any $g \in L^2(U)$, write $g = \sum_{k=1}^\infty c_k e_k$ where $c_k = (g, e_k)_{L^2}$. The candidate semigroup is \begin{align*} T(t)g &:= \sum_{k=1}^\infty e^{-\lambda_k t} c_k e_k. \end{align*} This is well-defined in $L^2(U)$ for every $t \ge 0$, since $\|T(t)g\|_{L^2}^2 = \sum_{k=1}^\infty e^{-2\lambda_k t} c_k^2 \le \sum_{k=1}^\infty c_k^2 = \|g\|_{L^2}^2$. The family $\{T(t)\}_{t \ge 0}$ satisfies: - $T(0)g = g$ (take $t = 0$ in the series). - $T(s)T(t)g = T(s+t)g$ for all $s, t \ge 0$ (multiply the exponentials: $e^{-\lambda_k s} e^{-\lambda_k t} = e^{-\lambda_k(s+t)}$). - $\|T(t)g - g\|_{L^2}^2 = \sum_{k=1}^\infty (e^{-\lambda_k t} - 1)^2 c_k^2 \to 0$ as $t \to 0^+$ by dominated convergence (since $(e^{-\lambda_k t} - 1)^2 c_k^2 \le 4c_k^2$ and $\sum c_k^2 < \infty$). The family $\{T(t)\}_{t \ge 0}$ is a $C_0$-semigroup on $L^2(U)$, and its generator is precisely $A = \Delta$ with domain $D(A) = H^2(U) \cap H^1_0(U)$. [/example] ## Definition The heat semigroup example reveals the algebraic and topological structure we want to axiomatize. We require: a consistent composition law (time additivity), a mild continuity at $t = 0$, and boundedness for each $t$. [definition: Strongly Continuous One-Parameter Semigroup] Let $X$ be a Banach space. A **strongly continuous one-parameter semigroup** (or **$C_0$-semigroup**) on $X$ is a family of bounded linear operators $\{T(t)\}_{t \ge 0} \subset \mathcal{L}(X)$ satisfying: 1. (**Identity**) $T(0) = I$, the identity operator on $X$. 2. (**Semigroup law**) $T(s + t) = T(s) T(t)$ for all $s, t \ge 0$. 3. (**Strong continuity**) For every $x \in X$, the map \begin{align*} [0, \infty) &\to X \\ t &\mapsto T(t)x \end{align*} is continuous. [/definition] The adjective "strongly continuous" refers to the topology on $X$, not on $\mathcal{L}(X)$. We require $\|T(t)x - T(0)x\|_X \to 0$ as $t \to 0^+$ for each fixed $x \in X$, but we do NOT require $\|T(t) - I\|_{\mathcal{L}(X)} \to 0$. This is a crucial distinction: differential operators like $\Delta$ are unbounded, and the operators $T(t) = e^{t\Delta}$ do not converge to $I$ in the operator norm as $t \to 0^+$. Strong continuity is the right assumption. [remark: Uniform vs. Strong Continuity] If instead $\|T(t) - I\|_{\mathcal{L}(X)} \to 0$ as $t \to 0^+$, the semigroup is called **uniformly continuous**. By a theorem of Phillips, this happens if and only if the generator $A$ is a bounded operator — which rules out all differential operators. Every uniformly continuous semigroup is $C_0$, but the converse fails dramatically. [/remark] The next step is to extract the "infinitesimal" information from the semigroup — the rate of change at $t = 0$. This is the generator, and it is the bridge back to the PDE. [definition: Infinitesimal Generator] Let $\{T(t)\}_{t \ge 0}$ be a $C_0$-semigroup on a Banach space $X$. The **infinitesimal generator** of the semigroup is the operator $A: D(A) \subset X \to X$ defined by \begin{align*} Ax &:= \lim_{t \to 0^+} \frac{T(t)x - x}{t}, \end{align*} where the **domain** $D(A)$ consists of all $x \in X$ for which this limit exists in the norm topology of $X$. [/definition] The domain $D(A)$ is not all of $X$ — this is what makes the theory nontrivial. For the heat semigroup, $D(A) = H^2(U) \cap H^1_0(U)$, a proper dense subset of $L^2(U)$. The generator is always a closed, densely defined operator, as the next fundamental result records. [quotetheorem:3144] The third part of this theorem is the abstract version of the PDE: the curve $t \mapsto T(t)x$ satisfies the abstract Cauchy problem $u'(t) = Au(t)$ in $X$. [definition: Growth Bound] Let $\{T(t)\}_{t \ge 0}$ be a $C_0$-semigroup on $X$. The **growth bound** (or **type**) of the semigroup is \begin{align*} \omega_0 &:= \inf\{\omega \in \mathbb{R} : \text{there exists } M \ge 1 \text{ such that } \|T(t)\|_{\mathcal{L}(X)} \le M e^{\omega t} \text{ for all } t \ge 0\}. \end{align*} A semigroup is called a **contraction semigroup** if $\|T(t)\|_{\mathcal{L}(X)} \le 1$ for all $t \ge 0$, i.e., $\omega_0 \le 0$ with $M = 1$. [/definition] Every $C_0$-semigroup satisfies an exponential bound: there exist constants $M \ge 1$ and $\omega \in \mathbb{R}$ such that $\|T(t)\|_{\mathcal{L}(X)} \le M e^{\omega t}$ for all $t \ge 0$. This follows from strong continuity alone, by a uniform boundedness argument. The growth bound $\omega_0$ is the smallest such $\omega$ achievable (possibly with varying $M$). [example: A Semigroup That Is Not a Contraction] On $X = L^2(0, \infty)$, define the **left shift semigroup** by \begin{align*} (T(t)f)(s) &:= f(s + t), \quad s \ge 0, \ t \ge 0. \end{align*} For each $t \ge 0$, $T(t): L^2(0, \infty) \to L^2(0, \infty)$ is bounded with $\|T(t)f\|_{L^2}^2 = \int_0^\infty |f(s+t)|^2 \, d\mathcal{L}^1(s) = \int_t^\infty |f(r)|^2 \, d\mathcal{L}^1(r) \le \|f\|_{L^2}^2$. So $\|T(t)\|_{\mathcal{L}(L^2)} \le 1$ and this is a contraction semigroup. By contrast, on $X = L^2(\mathbb{R})$, the **multiplication semigroup** $(T(t)f)(x) := e^{tx} f(x)$ gives $\|T(t)f\|_{L^2}^2 = \int_{\mathbb{R}} e^{2tx} |f(x)|^2 \, d\mathcal{L}^1(x)$. To see that $\|T(t)\|_{\mathcal{L}(L^2)} = \infty$ for each $t > 0$, take the test family $f_R := \mathbf{1}_{[R, R+1]} \in L^2(\mathbb{R})$, which satisfies $\|f_R\|_{L^2}^2 = 1$ for each $R > 0$. Then \begin{align*} \|T(t)f_R\|_{L^2}^2 &= \int_R^{R+1} e^{2tx} \, d\mathcal{L}^1(x) = \frac{e^{2t(R+1)} - e^{2tR}}{2t} = \frac{e^{2tR}(e^{2t} - 1)}{2t} \to \infty \end{align*} as $R \to \infty$, for any fixed $t > 0$. Since $\|f_R\|_{L^2} = 1$ and $\|T(t)f_R\|_{L^2} \to \infty$, we conclude $\|T(t)\|_{\mathcal{L}(L^2)} = \infty$: the semigroup is not even bounded as an operator on $L^2(\mathbb{R})$. The issue is that the generator $A = $ multiplication by $x$ is not even the right type of operator — the domain $D(A) = \{f \in L^2(\mathbb{R}) : xf \in L^2(\mathbb{R})\}$ does not lead to a valid semigroup on all of $L^2(\mathbb{R})$. [/example] ## The Hille-Yosida Theorem The central question of the abstract theory is: given a closed, densely defined operator $A: D(A) \subset X \to X$, when does $A$ generate a $C_0$-semigroup? A naive attempt — "define $e^{tA}$ by the exponential power series" — fails immediately: the series $\sum_{k=0}^\infty t^k A^k / k!$ requires iterating $A$ and controlling the resulting norms, but for an unbounded operator these norms grow without bound. A more sophisticated approach replaces $A$ by bounded approximations and passes to a limit. The key idea is to use the **resolvent** of $A$. For a complex number $\lambda$ with $\operatorname{Re}(\lambda) > \omega_0$, the operator $\lambda I - A$ should be invertible, and its inverse $R(\lambda, A) := (\lambda I - A)^{-1} \in \mathcal{L}(X)$ is the resolvent. The Yosida approximant replaces $A$ by the bounded operator $A_\lambda := \lambda A R(\lambda, A) = \lambda^2 R(\lambda, A) - \lambda I$. [remark: The Yosida Approximation Strategy] The bounded operators $A_\lambda$ defined above each generate a uniformly continuous semigroup $e^{tA_\lambda}$ via the standard power series, since $A_\lambda \in \mathcal{L}(X)$. The Hille-Yosida theorem is proved by establishing that as $\lambda \to \infty$, one has $A_\lambda x \to Ax$ for $x \in D(A)$, and consequently the semigroups $e^{tA_\lambda}$ converge strongly to a limit $T(t)$ in $\mathcal{L}(X)$. This limit is the desired semigroup generated by $A$. The resolvent condition in the theorem below is precisely what guarantees the necessary uniform estimates on $\|e^{tA_\lambda}\|_{\mathcal{L}(X)}$ as $\lambda \to \infty$. [/remark] [definition: Resolvent Set and Resolvent Operator] Let $A: D(A) \subset X \to X$ be a closed, densely defined operator on a Banach space $X$. The **resolvent set** of $A$ is \begin{align*} \rho(A) &:= \{\lambda \in \mathbb{C} : \lambda I - A: D(A) \to X \text{ is a bijection with bounded inverse}\}. \end{align*} For $\lambda \in \rho(A)$, the **resolvent operator** is $R(\lambda, A) := (\lambda I - A)^{-1} \in \mathcal{L}(X)$. [/definition] [explanation: Why the Resolvent Controls the Semigroup] The connection between the resolvent and the semigroup is not accidental. If $\{T(t)\}_{t \ge 0}$ is a semigroup with growth bound $\omega_0 < \lambda$, one can form the Laplace transform \begin{align*} \int_0^\infty e^{-\lambda t} T(t)x \, d\mathcal{L}^1(t), \end{align*} which converges in $X$ for $\lambda > \omega_0$. This Laplace transform turns out to equal $R(\lambda, A)x$. In other words, the resolvent IS the Laplace transform of the semigroup, and one can in principle recover the semigroup from the resolvent by the inverse Laplace transform. The Hille-Yosida theorem makes this circle of ideas rigorous: the resolvent conditions it imposes are precisely what is needed for the inverse Laplace transform to produce a valid semigroup. This perspective also explains why the conditions in Hille-Yosida involve the resolvent for all large real $\lambda$ (on the real axis to the right of the spectrum), rather than for complex $\lambda$. The Laplace transform for real $\lambda > \omega_0$ already determines the semigroup for $t > 0$. [/explanation] Now we can state the main theorem. It characterizes generators of contraction semigroups; a more general version handles arbitrary growth bounds. [quotetheorem:3139] The condition $\|R(\lambda, A)\|_{\mathcal{L}(X)} \le 1/\lambda$ for contraction semigroups is called the **Hille-Yosida condition**. The stronger condition involving powers of the resolvent is sometimes called the **Feller-Miyadera-Phillips condition**. [explanation: What the Resolvent Condition Means Geometrically] The estimate $\|R(\lambda, A)\|_{\mathcal{L}(X)} \le 1/\lambda$ says that the resolvent at $\lambda$ decays like $1/\lambda$ as $\lambda \to \infty$. Heuristically, $R(\lambda, A) \approx A^{-1}$ for large $\lambda$ would be wrong — rather, for large $\lambda$ the term $\lambda I$ dominates $A$, so $(\lambda I - A)^{-1} \approx \lambda^{-1} I$, giving $\|R(\lambda, A)\| \approx \lambda^{-1}$. The condition ensures this approximation holds with a correct uniform bound, which is exactly what is needed for the Yosida approximants $A_\lambda = \lambda^2 R(\lambda, A) - \lambda I$ to generate uniformly bounded semigroups that converge as $\lambda \to \infty$. For Hilbert spaces, there is a simpler criterion due to Lumer and Phillips: $A$ generates a contraction semigroup if and only if $A$ is **dissipative** (meaning $\operatorname{Re}(Ax, x)_H \le 0$ for all $x \in D(A)$) and the range of $\lambda I - A$ is all of $X$ for some (hence all) $\lambda > 0$. This Lumer-Phillips theorem is often easier to verify in practice for PDE operators. [/explanation] [quotetheorem:3140] [example: The Laplacian Generates a Contraction Semigroup] Let $X = L^2(U)$ with inner product $(f, g)_{L^2} = \int_U f \bar{g} \, d\mathcal{L}^n$, and let $A = \Delta$ with $D(A) = H^2(U) \cap H^1_0(U)$. We verify the Lumer-Phillips conditions. **Dissipativity:** For $u \in D(A)$, integrate by parts using Green's first identity. Since $u = 0$ on $\partial U$, the boundary terms vanish: \begin{align*} (\Delta u, u)_{L^2} &= \int_U (\Delta u) \bar{u} \, d\mathcal{L}^n = -\int_U \nabla u \cdot \nabla \bar{u} \, d\mathcal{L}^n = -\|\nabla u\|_{L^2}^2 \le 0. \end{align*} Since the inner product is real-valued for real $u$, we have $\operatorname{Re}(\Delta u, u)_{L^2} = -\|\nabla u\|_{L^2}^2 \le 0$, so $\Delta$ is dissipative. **Surjectivity:** For $\lambda > 0$ and $f \in L^2(U)$, we need to solve $(\lambda I - \Delta)u = f$, i.e., find $u \in D(A)$ satisfying $\lambda u - \Delta u = f$. This is the elliptic problem \begin{align*} -\Delta u + \lambda u &= f \quad \text{in } U \\ u &= 0 \quad \text{on } \partial U. \end{align*} The associated bilinear form $B[u,v] = \int_U (\nabla u \cdot \nabla \bar{v} + \lambda u \bar{v}) \, d\mathcal{L}^n$ is coercive on $H^1_0(U)$: $B[u, u] = \|\nabla u\|_{L^2}^2 + \lambda \|u\|_{L^2}^2 \ge \min(1, \lambda) \|u\|_{H^1}^2$. By the Lax-Milgram theorem, there exists a unique $u \in H^1_0(U)$ with $B[u, v] = (f, v)_{L^2}$ for all $v \in H^1_0(U)$, and elliptic regularity gives $u \in H^2(U)$. So $\lambda I - \Delta$ is surjective. By Lumer-Phillips, $\Delta$ generates a contraction semigroup on $L^2(U)$. This is the rigorous foundation for the heat semigroup discussed at the outset. [/example] ## The Abstract Cauchy Problem With the Hille-Yosida theorem in hand, we can now give a precise meaning to the abstract evolution equation that a $C_0$-semigroup is supposed to solve. The question is: given $A$ generating a semigroup $\{T(t)\}_{t \ge 0}$ and an initial datum $g \in X$, what is the solution theory for $u'(t) = Au(t)$, $u(0) = g$? [definition: Classical Solution of the Abstract Cauchy Problem] Let $A: D(A) \subset X \to X$ generate a $C_0$-semigroup $\{T(t)\}_{t \ge 0}$ on a Banach space $X$. Given $g \in X$ and $T > 0$, a **classical solution** of the abstract Cauchy problem \begin{align*} u'(t) &= Au(t), \quad t \in (0, T) \\ u(0) &= g \end{align*} is a function $u \in C^1([0, T]; X) \cap C([0, T]; D(A))$ (where $D(A)$ carries the graph norm $\|x\|_{D(A)} := \|x\|_X + \|Ax\|_X$) satisfying the equation pointwise. [/definition] [quotetheorem:3145] The distinction between classical and mild solutions is important for PDEs. In the heat equation context: if the initial data $g \in H^2(U) \cap H^1_0(U) = D(A)$, then $u(t) = T(t)g$ is a genuine classical solution of $\partial_t u = \Delta u$. If merely $g \in L^2(U)$, then $u(t) = T(t)g$ is the mild solution — it satisfies the equation in a weaker integrated sense, not pointwise. [example: What Happens Without the Domain Condition] Consider the heat equation with initial data $g(x) = \mathbb{1}_{[1/3, 2/3]}(x)$ on $U = (0,1) \subset \mathbb{R}$, viewed as an element of $L^2(0,1)$. Since $g$ is not even continuous, certainly $g \notin H^2(0,1)$, so $g \notin D(A)$. The mild solution $u(t) = T(t)g$ is well-defined as an element of $L^2(0,1)$ for each $t \ge 0$, and $t \mapsto u(t)$ is continuous in $L^2$. However, trying to compute $u'(0) = Au(0) = \Delta g$ makes no sense: the distributional Laplacian $\Delta g$ involves delta measures at $x = 1/3$ and $x = 2/3$ (the jump points of $g$), which lie outside $L^2$. The formula $u'(0) = Au(0)$ literally does not hold in $L^2$, confirming that we only have a mild solution, not a classical one. For any $t > 0$, however, $u(t) = T(t)g$ is smooth (analytic, in fact — see the section on analytic semigroups below), so the solution instantaneously regularizes. [/example] ## Duhamel's Principle for Inhomogeneous Problems The abstract Cauchy problem becomes richer and more applicable when a forcing term is present. The inhomogeneous abstract evolution equation \begin{align*} u'(t) &= Au(t) + f(t), \quad t \in (0, T) \\ u(0) &= g \end{align*} models, for instance, the heat equation with a heat source $f(x, t)$, or a wave equation driven by an external force. The solution formula generalizes the classical variation-of-parameters formula for ODEs. [quotetheorem:3141] The formula $u(t) = T(t)g + \int_0^t T(t-s)f(s) \, d\mathcal{L}^1(s)$ is the **Duhamel formula** or **variation-of-constants formula** in the abstract setting. The first term $T(t)g$ propagates the initial data forward in time, while the integral accumulates the contributions from the forcing term $f(s)$ at each earlier time $s$, propagating each contribution forward for a duration of $t - s$. [explanation: Deriving Duhamel's Formula Heuristically] The derivation mirrors the classical ODE proof. Multiply the equation $u'(t) = Au(t) + f(t)$ by the "integrating factor" $T(-t)$ (a heuristic — $T(-t)$ does not exist in general, as the semigroup runs only forward). Instead, differentiate $T(t - s)u(s)$ with respect to $s$ for $s \in [0, t]$: \begin{align*} \frac{d}{ds}\bigl[T(t-s)u(s)\bigr] &= -AT(t-s)u(s) + T(t-s)u'(s) \\ &= T(t-s)\bigl[u'(s) - Au(s)\bigr] = T(t-s)f(s). \end{align*} Integrating $s$ from $0$ to $t$ and using the fundamental theorem of calculus: \begin{align*} T(0)u(t) - T(t)u(0) &= \int_0^t T(t-s)f(s) \, d\mathcal{L}^1(s). \end{align*} Since $T(0) = I$ and $u(0) = g$, this yields $u(t) = T(t)g + \int_0^t T(t-s)f(s) \, d\mathcal{L}^1(s)$. The above derivation is purely formal — it treats $T(t-s)u(s)$ as differentiable in $s$, which requires $u(s) \in D(A)$ for each $s$. The rigorous proof proceeds by verifying that the formula satisfies the mild solution integral equation, and then showing classical regularity under stronger hypotheses on $g$ and $f$. [/explanation] [example: The Heat Equation with Source Term] Consider $X = L^2(U)$, $A = \Delta$ with $D(A) = H^2(U) \cap H^1_0(U)$, and the heat semigroup $\{T(t)\}_{t \ge 0}$ with eigenfunction expansion $T(t)h = \sum_{k=1}^\infty e^{-\lambda_k t}(h, e_k)_{L^2} e_k$. Take $g = 0$ and the scalar-separated source $f(\cdot, t) = \phi(t) e_1(\cdot)$ where $\phi \in L^1(0, T)$ is a given scalar function and $e_1$ is the first Dirichlet eigenfunction of $-\Delta$. The Duhamel formula gives \begin{align*} u(t) &= \int_0^t T(t - s)\bigl[\phi(s) e_1\bigr] \, d\mathcal{L}^1(s) = \int_0^t \phi(s) e^{-\lambda_1(t-s)} e_1 \, d\mathcal{L}^1(s). \end{align*} Since $e_1$ is independent of $s$, this is \begin{align*} u(t) &= \left(\int_0^t e^{-\lambda_1(t-s)} \phi(s) \, d\mathcal{L}^1(s)\right) e_1. \end{align*} The solution is a scalar convolution of $\phi$ with the exponential kernel $e^{-\lambda_1 \cdot}$, multiplied by the eigenfunction $e_1$. In particular, $\|u(t)\|_{L^2} = \left|\int_0^t e^{-\lambda_1(t-s)}\phi(s) \, d\mathcal{L}^1(s)\right|$ (since $\|e_1\|_{L^2} = 1$). This explicit formula shows that the source $\phi$ excites only the first eigenmode, and the response decays at rate $\lambda_1$ between source pulses — precisely the parabolic filtering behavior that Duhamel encodes. [/example] ## Applications to Parabolic and Hyperbolic Equations The abstract framework applies uniformly to very different types of evolution equations, revealing a structural distinction between parabolic and hyperbolic problems. The key difference lies in the spectral properties of the generator and the decay rate of the semigroup. ### Parabolic Equations For the heat equation and more general second-order parabolic equations, the generator $A$ is a second-order elliptic operator. Its spectrum lies in a left half-plane or sector of the complex plane, and the semigroup decays exponentially. Moreover, parabolic semigroups exhibit an **instantaneous regularization** property: even for rough initial data $g \in L^2$, the solution $u(t) = T(t)g$ becomes smooth for any $t > 0$. [quotetheorem:3142] The regularization follows from the spectral representation: $T(t)g = \sum_{k=1}^\infty e^{-\lambda_k t} c_k e_k$, and the factor $e^{-\lambda_k t}$ decays faster than any polynomial in $\lambda_k$ for $t > 0$. Since $\lambda_k \sim k^{2/n}$ by Weyl's law, the sum converges in all Sobolev norms. Quantitatively, applying $(-\Delta)^m$ (for integer $m$) introduces factors $\lambda_k^m$ in the series, and $\sup_k \lambda_k^m e^{-\lambda_k t} \le (m/t)^m e^{-m}$ by optimizing $\lambda \mapsto \lambda^m e^{-\lambda t}$, which gives $\|(-\Delta)^m T(t)g\|_{L^2} \le (m/t)^m e^{-m} \|g\|_{L^2}$. This is the sense in which $C_{k,t} = O(t^{-k})$ as $t \to 0^+$: the regularization comes at the cost of a polynomial singularity in $t$ near zero. ### Hyperbolic Equations For the wave equation, the situation is entirely different. Write the second-order wave equation $\partial_{tt} u - \Delta u = 0$ as a first-order system by introducing $v = \partial_t u$: \begin{align*} \frac{d}{dt} \begin{pmatrix} u \\ v \end{pmatrix} &= \begin{pmatrix} 0 & I \\ \Delta & 0 \end{pmatrix} \begin{pmatrix} u \\ v \end{pmatrix}. \end{align*} The generator is the operator $A = \begin{pmatrix} 0 & I \\ \Delta & 0 \end{pmatrix}$ on the energy space $X = H^1_0(U) \times L^2(U)$, with domain $D(A) = (H^2(U) \cap H^1_0(U)) \times H^1_0(U)$. The energy norm on $X$ is $\|(u, v)\|_X^2 = \|\nabla u\|_{L^2}^2 + \|v\|_{L^2}^2$. [quotetheorem:3146] [explanation: Why Hyperbolic Problems Can Be Reversed in Time] The fundamental difference between parabolic and hyperbolic equations is **time reversibility**. For the heat equation, time reversal gives the backward heat equation $\partial_t u = -\Delta u$ for $t > 0$, which is ill-posed: small perturbations to final data explode backward in time. This corresponds to the fact that the heat semigroup is not a group — $T(t)$ has no inverse for $t > 0$ (it annihilates high-frequency modes exponentially). For the wave equation, the physics is time-reversible: the generator $A$ generates a $C_0$-group, meaning $T(t)$ is invertible for every $t \in \mathbb{R}$ (with $T(t)^{-1} = T(-t)$). The energy is preserved, so no information is lost. Mathematically, this comes from the skew-adjointness of $A$ on $X$: one can check that $A^* = -A$, which by Stone's theorem (the group analog of Lumer-Phillips) characterizes generators of unitary groups on Hilbert spaces. [/explanation] [example: Wave Equation Energy Is Conserved] Consider $U = (0, \pi)$ in one dimension, with the wave equation $\partial_{tt} u = \partial_{xx} u$, $u(0,t) = u(\pi, t) = 0$, and initial data $(u(\cdot, 0), \partial_t u(\cdot, 0)) = (u_0, u_1) \in H^1_0(0, \pi) \times L^2(0, \pi)$. The eigenfunctions of $-\partial_{xx}$ with Dirichlet conditions are $e_k(x) = \sqrt{2/\pi}\sin(kx)$ with eigenvalues $\lambda_k = k^2$. Write $u_0 = \sum_{k=1}^\infty a_k e_k$ and $u_1 = \sum_{k=1}^\infty b_k e_k$. The solution is \begin{align*} u(x, t) &= \sum_{k=1}^\infty \left(a_k \cos(kt) + \frac{b_k}{k}\sin(kt)\right) e_k(x). \end{align*} The energy at time $t$ is \begin{align*} \|(u(\cdot, t), \partial_t u(\cdot, t))\|_X^2 &= \|\partial_x u(\cdot, t)\|_{L^2}^2 + \|\partial_t u(\cdot, t)\|_{L^2}^2. \end{align*} Computing term by term using Parseval: \begin{align*} \|\partial_x u(\cdot, t)\|_{L^2}^2 &= \sum_{k=1}^\infty k^2 \left(a_k \cos(kt) + \frac{b_k}{k}\sin(kt)\right)^2 \\ \|\partial_t u(\cdot, t)\|_{L^2}^2 &= \sum_{k=1}^\infty \left(-k a_k \sin(kt) + b_k \cos(kt)\right)^2. \end{align*} Adding these and using $\cos^2 + \sin^2 = 1$: \begin{align*} \|(u(\cdot, t), \partial_t u(\cdot, t))\|_X^2 &= \sum_{k=1}^\infty (k^2 a_k^2 + b_k^2) = \|(u_0, u_1)\|_X^2. \end{align*} The energy is exactly conserved — confirming that the wave semigroup is isometric. [/example] ## Analytic Semigroups Not all parabolic problems exhibit the same degree of regularity. For equations whose generator has good spectral properties in the complex plane — specifically, when the spectrum lies in a sector and the resolvent decays along rays in that sector — the semigroup extends to a holomorphic function on a sector of the complex plane. These are the **analytic semigroups**, and they provide even stronger regularity than the general $C_0$ theory. The key observation is that for the heat semigroup $T(t) = e^{t\Delta}$, the formula $\sum_{k=1}^\infty e^{-\lambda_k t} c_k e_k$ makes sense not just for real $t > 0$ but for any complex $t$ with $\operatorname{Re}(t) > 0$, since $|e^{-\lambda_k t}| = e^{-\lambda_k \operatorname{Re}(t)}$ decays exponentially as $k \to \infty$. So the heat semigroup extends to a holomorphic map from the right half-plane into $\mathcal{L}(L^2(U))$. <!-- illustration-needed: A sector $\Sigma_\delta = \{z \in \mathbb{C} : |\arg z| < \delta\}$ for $\delta \in (0, \pi/2)$, showing the real positive axis inside the sector, with the spectrum of $-A$ lying in the complementary closed sector on the left half-plane. --> [definition: Analytic Semigroup] Let $\delta \in (0, \pi/2)$ and let $\Sigma_\delta := \{z \in \mathbb{C} \setminus \{0\} : |\arg z| < \delta\}$ be the open sector of angle $\delta$. A $C_0$-semigroup $\{T(t)\}_{t \ge 0}$ on a Banach space $X$, where each $T(t) \in \mathcal{L}(X)$, is called an **analytic semigroup** (of angle $\delta$) if: 1. The map $t \mapsto T(t)$ extends to a holomorphic map $z \mapsto T(z)$ from $\Sigma_\delta$ into $\mathcal{L}(X)$. 2. $T(z_1 + z_2) = T(z_1) T(z_2)$ for all $z_1, z_2 \in \Sigma_\delta$. 3. $T(z)x \to x$ in $X$ as $z \to 0$ within any closed subsector $\overline{\Sigma_{\delta'}}$ with $\delta' < \delta$. [/definition] The analyticity of $z \mapsto T(z)$ in the complex sector has striking consequences. Differentiating with respect to $z$, we find that $T(z)x \in D(A^k)$ for all $x \in X$ and all $k \ge 1$: analytic semigroups instantly smooth all initial data into the domain of every power of $A$. For PDE applications, this translates to full spatial regularity. [quotetheorem:3143] [example: Second-Order Elliptic Operators Generate Analytic Semigroups] Let $L: H^2(U) \cap H^1_0(U) \to L^2(U)$ be the uniformly elliptic operator in divergence form on $U \subset \mathbb{R}^n$ defined by \begin{align*} Lu &:= -\sum_{i,j=1}^n \partial_{x_i}(a_{ij}\, \partial_{x_j} u) \end{align*} with coefficients $a_{ij} \in C^\infty(\bar{U})$ satisfying the uniform ellipticity condition $\sum_{i,j} a_{ij}(x)\xi_i\xi_j \ge \theta |\xi|^2$ for all $x \in U$, $\xi \in \mathbb{R}^n$, and some $\theta > 0$. Set $A = -L$ with domain $D(A) = H^2(U) \cap H^1_0(U)$ on $X = L^2(U)$. The spectrum of $A$ consists of eigenvalues $-\mu_k < 0$ with $\mu_k \to \infty$, lying on the negative real axis. The resolvent estimate holds in a sector: for $\lambda$ outside the sector $\{re^{i\alpha} : r \ge 0, |\alpha| \ge \pi/2 + \varepsilon\}$ for small $\varepsilon > 0$, a Lax-Milgram argument for complex $\lambda$ in this sector, combined with the coercivity of $L$, gives $\|R(\lambda, A)\|_{\mathcal{L}(L^2)} \le C/|\lambda|$. The semigroup $\{T(t)\}_{t \ge 0}$ generated by $A = -L$ is therefore analytic. For any $g \in L^2(U)$ and $t > 0$: $T(t)g \in D(A^k)$ for all $k$, which by elliptic regularity theory means $T(t)g \in H^{2k}(U)$ for all $k \ge 1$. In particular, $T(t)g \in C^\infty(\bar{U})$, and the solution of $\partial_t u + Lu = 0$ becomes smooth instantaneously. [/example] [remark: Fractional Powers of Generators] For analytic semigroups, one can define **fractional powers** $(-A)^\alpha$ for $\alpha \in (0, 1)$ via the formula $(-A)^{-\alpha} = \frac{1}{\Gamma(\alpha)}\int_0^\infty t^{\alpha-1} T(t) \, d\mathcal{L}^1(t)$. The domains $D((-A)^\alpha)$ form an interpolation scale between $X$ and $D(A)$, and they coincide with fractional-order Sobolev spaces in the PDE context. This framework underpins the theory of **abstract parabolic equations** as developed by Kato and Tanabe. [/remark] ## Spectral Mapping and Long-Time Behavior Understanding the long-time behavior of solutions to evolution equations — whether they decay, grow, or oscillate — reduces to understanding the spectrum of the generator. This is the **spectral mapping theorem** for semigroups, and it connects the spectrum of the semigroup operators $T(t)$ to the spectrum of the generator $A$. The naive hope is that $\sigma(T(t)) = e^{t\sigma(A)} := \{e^{t\lambda} : \lambda \in \sigma(A)\}$. For uniformly continuous semigroups (bounded generators), this holds exactly. For $C_0$-semigroups, one always has the inclusion \begin{align*} e^{t\sigma(A)} &\subset \sigma(T(t)) \setminus \{0\}, \end{align*} but the reverse inclusion can fail in general Banach spaces. For analytic semigroups, the full spectral mapping theorem holds: $\sigma(T(t)) \setminus \{0\} = e^{t\sigma(A)}$. [quotetheorem:3147] [explanation: Stability for the Heat Equation] For the heat equation on a bounded domain $U$, the eigenvalues of $\Delta$ with Dirichlet conditions are $\{-\lambda_k\}_{k=1}^\infty$ with $0 < \lambda_1 \le \lambda_2 \le \ldots$ and $\lambda_1 > 0$. The spectral bound is $s(A) = -\lambda_1 < 0$. Since the heat semigroup is analytic, $\omega_0 = s(A) = -\lambda_1 < 0$. This means \begin{align*} \|T(t)\|_{\mathcal{L}(L^2)} &\le M e^{-\lambda_1 t} \end{align*} for some $M \ge 1$. In fact $M = 1$ works (it is a contraction semigroup), so $\|T(t)g\|_{L^2} \le e^{-\lambda_1 t}\|g\|_{L^2}$. Every solution of the heat equation on a bounded domain decays exponentially in $L^2$, with the rate controlled by the first Dirichlet eigenvalue $\lambda_1$ of $-\Delta$. This is the rigorous version of the intuition that "heat diffuses and dissipates" in a bounded enclosure. On an unbounded domain such as $\mathbb{R}^n$, the Dirichlet spectrum is $[0, \infty)$ and $\lambda_1 = 0$, so there is no exponential decay in $L^2(\mathbb{R}^n)$ — indeed, the heat kernel mass is preserved and solutions spread out but do not contract. [/explanation] ## References - L.C. Evans, *Partial Differential Equations* (2010), Chapter 7. - A. Pazy, *Semigroups of Linear Operators and Applications to Partial Differential Equations* (1983). - K.-J. Engel and R. Nagel, *One-Parameter Semigroups for Linear Evolution Equations* (2000). - E. Hille and R.S. Phillips, *Functional Analysis and Semi-Groups* (1957). - H. Tanabe, *Equations of Evolution* (1979). - T. Kato, *Perturbation Theory for Linear Operators* (1966), Chapter IX.