the Cauchy problem for the heat equation
/ KOH-shee /
Suppose you know the temperature everywhere along an infinitely long wire at one instant, and there are no ends to worry about. Can you predict the temperature for all later times? The Cauchy problem for the heat equation is exactly this clean question: given the starting profile on the whole line, find the future.
Formally, the Cauchy problem is: solve u_t = k u_xx for all real x and t > 0, subject only to the initial condition u(x,0) = f(x). There are no boundary conditions because there is no boundary — the domain is the entire line (or all of space). The solution is given by convolving the initial data with the heat kernel: u(x,t) = integral over all y of G(x - y, t) f(y) dy, where G is the Gaussian heat kernel. You can read this as: the future temperature at x is a Gaussian-weighted average of the initial temperatures, blending nearby values heavily and distant values lightly. As t goes to zero this average collapses back onto f, recovering the initial condition; as t grows the averaging window widens and everything blurs together.
This is the model for an initial value problem with no spatial boundary, and it makes the heat equation's character vivid. The convolution formula instantly shows infinite propagation speed (every point of f influences every point of u for any t > 0), smoothing (G is infinitely smooth, so u is too, however rough f was), and conservation of total heat (the integral of u stays equal to the integral of f). Honest caveats: forward in time the problem is well-posed; the backward Cauchy problem is ill-posed. And uniqueness on the line needs a mild growth restriction on u — without it Tychonoff found bizarre nonzero solutions with zero initial data, so 'reasonable' growth is part of the well-posed statement.
If f is a single rectangular block of heat, the solution u(x,t) is that block blurred by the Gaussian — its sharp corners round off instantly and it spreads into a smooth hump.
Convolution with the kernel = instant blurring of the initial data.
Uniqueness on the unbounded line requires a growth condition (e.g. u not growing faster than e^(a x^2)); drop it and uniqueness fails. The Cauchy problem is well-posed forward in time only — never backward.