a Bochner space
/ BOKH-ner /
When you study an evolution equation, the cleanest way to think about the solution u(x, t) is to forget the x for a moment and watch a single moving point: at each time t you hold a whole spatial profile u(., t) — a function of x — and as t advances that profile drifts through a space of functions. The solution becomes a curve, t goes to u(t), tracing a path through an infinite-dimensional space. A Bochner space is the natural home for such curves: it measures functions of time whose values are themselves elements of a Banach space (often a Sobolev space).
Concretely, given a Banach space X (think of X = L^2(Omega) or the Sobolev space H^1_0(Omega) of spatial profiles), the Bochner space L^p(0, T; X) consists of maps u from the time interval (0, T) into X for which the number integral from 0 to T of (norm of u(t) in X)^p dt is finite — exactly the ordinary L^p definition, but with the absolute value replaced by the X-norm. So u belongs to L^2(0, T; H^1_0) means: at almost every instant the profile lives in H^1_0, and its H^1-size, squared and integrated over time, is finite. One writes the weak formulation of a parabolic problem in precisely these terms: typically u is in L^2(0, T; H^1_0) with time derivative u' in L^2(0, T; H^{-1}).
This bookkeeping is what makes the modern existence theory work. Galerkin approximation builds solutions in finite-dimensional spatial subspaces; the energy estimates you prove are bounds in Bochner norms; and a compactness argument passes to the limit. Splitting 'time' and 'space' this way — an ordinary integral in t over a Banach-valued integrand — turns a PDE in (x, t) into something that reads almost like an ODE u' + A u = f in the abstract space X, which is the doorway to the semigroup viewpoint.
For the heat IBVP, the standard solution class is u in L^2(0, T; H^1_0(Omega)) with u' in L^2(0, T; H^{-1}(Omega)). The first says 'the spatial profile is in H^1_0 most of the time, with finite total H^1 energy'; the second says 'the rate of change, measured in the weaker dual norm, is also square-integrable in time'. Together they make u continuous in time with values in L^2.
The natural function class in which a weak parabolic solution lives.
A Bochner space is not the same as a space of functions of (x, t) jointly: L^2(0, T; H^1_0) controls the spatial H^1-norm at each time and integrates that in t, which is genuinely stronger than just being L^2 in (x, t). Keeping the order 'integrate the X-norm over time' straight is the whole point.