摘要
We look for solutions u (x, t) of the one-dimensional heat equation ut = uxx which are space-time periodic, i.e. they satisfy the property u (x + a, t + b) = u (x, t) for all (x, t) ∈ (−∞, ∞) × (−∞, ∞) , and derive their Fourier series expansions. Here a ≥ 0, b ≥ 0 are two constants with a + b > 0. For general equation of the form ut = uxx + Aux + Bu, where A, B are two constants, we also have similar results. Moreover, we show that non-constant bounded periodic solution can occur only when B > 0 and is given by a linear combination of cos ( √B (x + At) ) and sin ( √B (x + At) ) .