independent and dependent variables
Every PDE has two kinds of quantity, and keeping them straight is the first skill you need. The independent variables are the inputs you are free to dial — typically the spatial positions x, y, z and the time t. The dependent variable is the output that responds — the unknown function u, whose value u(x, y, z, t) is determined once you fix where and when you are looking. In a weather model, latitude, longitude, altitude and time are independent; the temperature, pressure and wind speed at that point are dependent.
The reason the distinction is sharp is that we differentiate only with respect to the independent variables. When you write u_x you mean 'how does u change as I move in x, holding y, z, t fixed' — so u_x, u_t, u_xx are all derivatives of the dependent variable with respect to independent ones. The number of independent variables sets the 'dimension' of the problem: an equation in x and t alone (one space, one time) is much simpler to picture than one in x, y, z and t. The dependent variable can itself be a single number (a scalar, like temperature) or a list of numbers (a vector, like a velocity field with three components), and that choice decides whether you have a single equation or a system.
Naming the variables clearly is not bookkeeping for its own sake — it tells you what kind of conditions the problem will need. Time t usually calls for an initial condition (the state at t = 0), while space variables call for boundary conditions (what happens at the edges of the region). Mixing them up is a classic beginner error: treating a spatial endpoint as if it were an initial time, or vice versa, can turn a sensible problem into a meaningless one.
In u_t = k u_xx the independent variables are x (position) and t (time); the dependent variable is u (temperature). We supply one initial condition u(x, 0) = f(x) in time and two boundary conditions, say u(0, t) and u(L, t), in space.
One space variable, one time variable, one scalar unknown — the simplest non-trivial setting.
Time and space are both independent variables, but they are not interchangeable: time usually enters through one derivative (or two, for waves) and gets an initial condition, while space enters through second derivatives and gets boundary conditions. The asymmetry is what makes evolution problems and steady-state problems behave so differently.