the abstract Cauchy problem
Here is a powerful trick of perspective. A diffusion equation u_t = -L u in (x, t) looks two-dimensional, but if you treat the whole spatial profile at a single time as one point in a function space, then the only thing actually changing is that point, and it changes with time. The equation collapses into something that reads like the simplest ODE in the world, du/dt = -A u — except u(t) is not a number, it is a vector in an infinite-dimensional space, and A is an operator, not a number. That recast is the abstract Cauchy problem.
Precisely: choose a Banach space X (for the heat equation X = L^2(Omega)) and a linear operator A on X (for the heat equation A = -Laplacian with its boundary conditions, which is unbounded — defined only on smooth-enough functions). The abstract Cauchy problem is: find a function u: [0, infinity) into X with du/dt = -A u for t > 0 and u(0) = u_0, where u_0 in X is the given initial profile. 'Cauchy problem' just means initial-value problem; 'abstract' means we no longer mention x, derivatives in space, or boundary conditions — all of that has been folded into the single operator A and its domain. Solving it means finding a curve through X whose velocity at every instant equals -A applied to where it currently is.
Why bother? Because once a concrete PDE is written this way, an entire ready-made machinery applies regardless of the original equation. The solution, when it exists, is u(t) = e^{-tA} u_0 — a semigroup of operators — and the central questions (does the curve exist, is it unique, does it stay bounded, does it decay) become questions about the single operator A: its spectrum, its resolvent, whether it generates a semigroup. The Hille-Yosida and Lumer-Phillips theorems answer exactly when the abstract Cauchy problem is well-posed.
The heat IBVP u_t = Laplacian u on Omega, u = 0 on the boundary, u(., 0) = u_0, becomes du/dt = -A u, u(0) = u_0, with X = L^2(Omega) and A = -Laplacian on the domain D(A) = H^2(Omega) intersect H^1_0(Omega). The boundary condition lives inside the choice of D(A), not in the abstract equation itself.
A PDE in (x, t) recast as an ODE in an infinite-dimensional space.
The operator A is almost always unbounded, so du/dt = -A u is not the elementary linear ODE it pretends to be — you cannot just write e^{-tA} u_0 as a power series and be done. Making sense of e^{-tA} for an unbounded A is exactly the content of semigroup theory, and the domain D(A) (where A is even defined) is not optional bookkeeping.