Semigroup theory treats an evolution equation as an ordinary differential equation in an infinite-dimensional space. The unbounded spatial operator is encoded by its generator, while the semigroup carries existence, uniqueness, regularity, and long-time behavior.
Strongly continuous semigroups
Definition. A \(C_0\)-semigroup on a Banach space \(X\) is a family \(T(t)\in\mathcal L(X)\), \(t\ge0\), such that \[ T(0)=I,\qquad T(t+s)=T(t)T(s),\qquad \lim_{t\downarrow0}T(t)x=x \] for every \(x\in X\). Its generator is \[ Ax=\lim_{t\downarrow0}\frac{T(t)x-x}{t}, \qquad D(A)=\left\{x:\text{the limit exists}\right\}. \]
Proposition. If \(A\in\mathcal L(X)\), then \[ e^{tA}=\sum_{k=0}^{\infty}\frac{t^kA^k}{k!} \] is a uniformly continuous semigroup with generator \(A\). Conversely, every uniformly continuous semigroup has a bounded generator and this form.
Differential operators are normally unbounded. Their domain is therefore part of the operator and cannot be omitted: the same formal expression with Dirichlet, Neumann, or periodic domains generates different dynamics.
Generation theorems
Theorem (Hille-Yosida, contraction case). A densely defined operator \(A\) generates a contraction \(C_0\)-semigroup if and only if \(A\) is closed, \((0,\infty)\subset\rho(A)\), and \[ \|(\lambda I-A)^{-1}\|\le\frac1\lambda \qquad(\lambda>0). \]
The resolvent estimates are the infinite-dimensional analogue of controlling \((\lambda I-A)^{-1}\) for a matrix. On a Hilbert space, the Lumer-Phillips theorem is often easier to apply: \(A\) generates a contraction semigroup exactly when \(A\) is maximal dissipative, \[ \operatorname{Re}\langle Ax,x\rangle\le0, \qquad \operatorname{Ran}(\lambda I-A)=X \] for some \(\lambda>0\).
Example (Dirichlet heat flow). On \(X=L^2(\Omega)\), let \(A=\Delta\) with \(D(A)=H^2(\Omega)\cap H_0^1(\Omega)\). Integration by parts gives \[ \langle Au,u\rangle=-\|\nabla u\|_2^2\le0. \] Lax-Milgram solves \((\lambda I-\Delta)u=f\), so \(A\) is maximal dissipative and generates the heat semigroup.
Classical and mild solutions
Theorem (variation of constants). If \(A\) generates \(T(t)\), the inhomogeneous problem \[ u'(t)=Au(t)+f(t),\qquad u(0)=u_0, \] has the mild solution \[ u(t)=T(t)u_0+\int_0^tT(t-s)f(s)\,\mathrm{d}s. \] If \(u_0\in D(A)\) and \(f\) has suitable regularity, this is a classical solution.
Proof. For a classical solution, differentiate \(T(t-s)u(s)\) with respect to \(s\): \[ \frac{\mathrm{d}}{\mathrm{d}s}\bigl(T(t-s)u(s)\bigr) =T(t-s)f(s). \] Integration from \(0\) to \(t\) yields the formula. For general data, the right-hand side defines the mild solution by density and continuity.
Analyticity and smoothing
A semigroup is analytic if \(T(z)\) extends holomorphically to a sector in the complex plane. Sectorial generators satisfy estimates such as \[ \|A^kT(t)\|\le C_k t^{-k},\qquad t>0, \] which express instantaneous regularization. Elliptic operators generate analytic parabolic semigroups; skew-adjoint operators generate unitary groups and describe conservative wave or Schrödinger evolution instead.
Nonlinear evolution
For \(u'=Au+F(u)\), the mild equation \[ u(t)=T(t)u_0+\int_0^tT(t-s)F(u(s))\,\mathrm{d}s \] is solved locally by contraction when \(F\) is locally Lipschitz. A priori estimates then decide continuation or blow-up. Maximal monotone and \(m\)-accretive operators extend the framework to nonlinear diffusion and subgradient flows where differentiability of \(F\) is unavailable.