a mild solution
What should you even mean by 'a solution' when the initial data is too rough to differentiate, or the operator is unbounded so the equation cannot literally hold at t = 0? The semigroup viewpoint offers a clean answer: declare the Duhamel integral formula itself to be the definition of a solution. A mild solution is a function that satisfies the variation-of-constants formula, full stop — it need not be differentiable, it need not lie in the domain of the operator at every instant. You give up demanding the equation hold pointwise and instead demand the integral identity.
Precisely, for du/dt = -A u + F(u), u(0) = u_0 with -A generating a semigroup e^{-tA}, a continuous function u: [0, T] into X is a mild solution if u(t) = e^{-tA} u_0 + integral from 0 to t of e^{-(t-s)A} F(u(s)) ds for all t in [0, T]. Compare three notions of solution, from strongest to weakest: a classical solution is differentiable in t, lives in D(A) for each t, and satisfies the equation literally; a strong solution is differentiable almost everywhere with the equation holding a.e.; a mild solution merely satisfies the integral formula. Every classical solution is mild, but a mild solution can be far less regular — for instance u_0 only continuous, no derivative anywhere.
Why work with the weakest notion? Because mild solutions are the ones you can actually construct. The integral formula turns the problem into a fixed point: define a map Phi(u)(t) = e^{-tA} u_0 + integral from 0 to t of e^{-(t-s)A} F(u(s)) ds and look for u with Phi(u) = u. On a short enough time interval Phi is a contraction (the smoothing of an analytic semigroup tames F), so the Banach fixed-point theorem hands you a unique mild solution — local existence for semilinear parabolic equations, with no a-priori smoothness assumed. Parabolic regularity theory then upgrades it: that same smoothing forces a mild solution to become classical for t > 0.
For a reaction-diffusion equation u_t = Laplacian u + u^2 with merely continuous initial data u_0, you first build a mild solution by iterating u_{n+1}(t) = e^{t Laplacian} u_0 + integral from 0 to t of e^{(t-s) Laplacian} u_n(s)^2 ds. The iteration converges on a short interval. Then parabolic smoothing shows this mild u is actually smooth for t > 0 and solves the PDE classically.
Build a mild solution by fixed point, then let parabolic smoothing make it classical.
A mild solution need not be unique or global merely by existing — for semilinear parabolic equations a mild solution can blow up in finite time even though it starts out fine (think u_t = Laplacian u + u^2). Existence as a mild solution is a local statement; whether it continues, stays bounded, or blows up is a separate question the energy and comparison methods address.