Let $A: D(A) \subset X \to X$ generate a $C_0$-semigroup $\{T(t)\}_{t \ge 0}$ on a real or complex Banach space $X$. Define the **growth bound** \begin{align*} \omega_0 &:= \inf\{\omega \in \mathbb{R} : \exists\, M_\omega \ge 1\ \text{such that}\ \|T(t)\|_{\mathcal{L}(X)} \le M_\omega e^{\omega t}\ \text{for all}\ t \ge 0\} \end{align*} and the **spectral bound** $s(A) := \sup\{\operatorname{Re}(\lambda) : \lambda \in \sigma(A)\}$. 1. Always $s(A) \le \omega_0$ (the spectral bound does not exceed the growth bound). 2. If $\{T(t)\}_{t \ge 0}$ is an analytic semigroup, then $s(A) = \omega_0$. 3. The semigroup satisfies $\|T(t)\|_{\mathcal{L}(X)} \le M e^{\omega t}$ for some $M \ge 1$ if and only if $\{\lambda \in \mathbb{C} : \operatorname{Re}(\lambda) > \omega\} \subset \rho(A)$ with…

Let $U \subset \mathbb{R}^n$ be a bounded open set with smooth boundary, and let $X = H^1_0(U) \times L^2(U)$ be equipped with the energy inner product \begin{align*} ((u_1, v_1), (u_2, v_2))_X &= \int_U \nabla u_1 \cdot \overline{\nabla u_2} \, d\mathcal{L}^n + \int_U v_1\, \overline{v_2} \, d\mathcal{L}^n, \end{align*} which is a Hilbert inner product because Poincaré's inequality makes $\|\nabla \cdot\|_{L^2}$ an equivalent norm to $\|\cdot\|_{H^1}$ on $H^1_0(U)$. Let $A: D(A) \to X$ be the wave operator \begin{align*} D(A) &= (H^2(U) \cap H^1_0(U)) \times H^1_0(U), & A\begin{pmatrix} u \\ v \end{pmatrix} &= \begin{pmatrix} v \\ \Delta u \end{pmatrix}. \end{align*} Then $A$ and $-A$ both generate $C_0$-semigroups of contractions on $X$. Together they form a **$C_0$-group**: there…

Let $A: D(A) \subset X \to X$ generate a $C_0$-semigroup $\{T(t)\}_{t \ge 0}$ on a real or complex Banach space $X$. The abstract Cauchy problem reads: find $u \in C^1([0, \infty); X) \cap C([0, \infty); D(A))$ with $u'(t) = A u(t)$ for $t \ge 0$ and $u(0) = g$. 1. If $g \in D(A)$, then $u(t) := T(t)g$ is the unique classical solution. Moreover, $u \in C^1([0, \infty); X) \cap C([0, \infty); D(A))$. 2. If $g \in X \setminus D(A)$, then $u(t) := T(t)g$ is continuous but generally not differentiable in $X$. It is called a **mild solution**.

Let $\{T(t)\}_{t \ge 0}$ be a $C_0$-semigroup on a real or complex Banach space $X$ with generator $A: D(A) \subset X \to X$. Then: 1. $D(A)$ is dense in $X$. 2. $A$ is a closed operator: if $x_n \in D(A)$, $x_n \to x$ in $X$, and $Ax_n \to y$ in $X$, then $x \in D(A)$ and $Ax = y$. 3. For $x \in D(A)$ and $t \ge 0$, we have $T(t)x \in D(A)$ and $\frac{d}{dt}T(t)x = AT(t)x = T(t)Ax$.

Let $A$ generate a $C_0$-semigroup $\{T(t)\}_{t \ge 0}$ on a Banach space $X$. The following are equivalent: 1. $\{T(t)\}_{t \ge 0}$ is an **analytic semigroup of angle $\delta \in (0, \pi/2)$**: it admits a holomorphic extension $T: \Sigma_\delta \to \mathcal{L}(X)$ on the open sector $\Sigma_\delta := \{z \in \mathbb{C} \setminus \{0\} : |\arg z| < \delta\}$ such that $T(z_1 + z_2) = T(z_1)T(z_2)$ for $z_1, z_2 \in \Sigma_\delta$, $T(z)$ agrees with the original semigroup on $(0, \infty)$, $T(z)x \to x$ as $z \to 0$ within every subsector $\Sigma_{\delta'}$ ($\delta' < \delta$), and $\|T(z)\|_{\mathcal{L}(X)}$ is bounded on every bounded subsector. 2. After a spectral shift $A \mapsto A - \omega I$ (which is harmless: it translates the spectrum and rescales the semigroup by $e^{-\omega t}$…

Let $U \subset \mathbb{R}^n$ be a bounded open set with smooth boundary. Let $A = \Delta$ with domain $D(A) = H^2(U) \cap H^1_0(U)$ on $X = L^2(U)$, generating the heat semigroup $\{T(t)\}_{t \ge 0}$. For any $g \in L^2(U)$ and $t > 0$: 1. $T(t)g \in H^k(U)$ for every $k \ge 0$. 2. $\|T(t)g\|_{H^k(U)} \le C_{k,t} \|g\|_{L^2(U)}$ for some constant $C_{k,t} > 0$ satisfying $C_{k,t} \le C_k \, t^{-k}$ as $t \to 0^+$ (for a constant $C_k$ depending only on $k$ and $U$). The conservative bound $t^{-k}$ is stated for simplicity; the proof in fact yields the sharper rate $C_{k,t} \le C_k\, t^{-k/2}$ as $t \to 0^+$. In particular, $T(t)g \in C^\infty(\bar{U})$ for $t > 0$ (and is real-analytic in the interior of $U$).

Let $A$ generate a $C_0$-semigroup $\{T(t)\}_{t \ge 0}$ on $X$. Let $g \in X$ and $f \in L^1(0, T; X)$. Consider the inhomogeneous abstract Cauchy problem \begin{align*} u'(t) &= Au(t) + f(t), \quad t \in (0, T), \\ u(0) &= g. \end{align*} The **mild solution** of this problem is \begin{align*} u(t) &= T(t)g + \int_0^t T(t - s) f(s) \, d\mathcal{L}^1(s), \quad t \in [0, T]. \end{align*} This is a well-defined element of $C([0, T]; X)$. If additionally $g \in D(A)$ and $f \in C^1([0, T]; X)$, then $u$ is a **classical solution**: that is, \begin{align*} u &\in C^1((0, T); X) \cap C([0, T]; X), \\ u(t) &\in D(A) \quad \text{for all } t \in (0, T), \\ u'(t) &= Au(t) + f(t) \quad \text{pointwise on } (0, T), \\ u(0) &= g. \end{align*}

Let $X$ be a Hilbert space with inner product $(\cdot, \cdot)_H$ and $A: D(A) \subset X \to X$ a densely defined operator. Then $A$ generates a $C_0$-semigroup of contractions if and only if: 1. $A$ is **dissipative**: $\operatorname{Re}(Ax, x)_H \le 0$ for all $x \in D(A)$. 2. There exists $\lambda > 0$ such that $\lambda I - A: D(A) \to X$ is surjective.

Let $X$ be a Banach space and $A: D(A) \subset X \to X$ a closed, densely defined linear operator. The following are equivalent: 1. $A$ generates a $C_0$-semigroup of contractions $\{T(t)\}_{t \ge 0}$ satisfying $\|T(t)\|_{\mathcal{L}(X)} \le 1$ for all $t \ge 0$. 2. Every $\lambda > 0$ belongs to the resolvent set $\rho(A)$, and the resolvent satisfies \begin{align*} \|R(\lambda, A)\|_{\mathcal{L}(X)} &\le \frac{1}{\lambda} \quad \text{for all } \lambda > 0. \end{align*} More generally, $A$ generates a $C_0$-semigroup satisfying $\|T(t)\|_{\mathcal{L}(X)} \le M e^{\omega t}$ for some $M \ge 1$ and $\omega \in \mathbb{R}$ if and only if every $\lambda > \omega$ belongs to $\rho(A)$ and \begin{align*} \|R(\lambda, A)^n\|_{\mathcal{L}(X)} &\le \frac{M}{(\lambda - \omega)^n} \quad \text{for all } \lambda > \omega, \ n \ge 1. \end{align*}…

Let $U \subset \mathbb{R}^n$ be a connected open set and let $L$ be a uniformly elliptic operator in divergence form: \begin{align*} Lu := -\sum_{i,j=1}^{n} \partial_{x_i}\!\bigl(a_{ij}(x)\,\partial_{x_j} u\bigr) + \sum_{i=1}^{n} b_i(x)\,\partial_{x_i} u + c(x)\,u, \end{align*} where the coefficients satisfy: 1. **Uniform ellipticity.** There exists $\theta > 0$ such that for all $x \in U$ and all $\xi \in \mathbb{R}^n$, \begin{align*} \sum_{i,j=1}^{n} a_{ij}(x)\,\xi_i\,\xi_j \ge \theta\,|\xi|^2. \end{align*} 2. **Boundedness.** The coefficients $a_{ij}, b_i, c$ belong to $L^\infty(U)$ with $\|a_{ij}\|_{L^\infty}, \|b_i\|_{L^\infty}, \|c\|_{L^\infty} \le \Lambda$ for some $\Lambda > 0$. 3. **Sign condition.** $c(x) \ge 0$ in $U$. Suppose $u \in H^1(U)$ is a weak solution of $Lu = 0$ in $U$…