2.3. HEAT EQUATION 55 Strong maximum principle for the heat equation 2.3.3. Properties of solutions. a. Strong maximum principle, uniqueness. First we employ the mean- value property to give a quick proof of the strong maximum principle. THEOREM 4 (Strong maximum principle for the heat equation). Assume u C1 2(UT ) C( ¯T U ) solves the heat equation in UT . (i) Then max ¯ U T u = max ΓT u. (ii) Furthermore, if U is connected and there exists a point (x0,t0) UT such that u(x0,t0) = max ¯ U T u, then u is constant in ¯ U t0 . Assertion (i) is the maximum principle for the heat equation and (ii) is the strong maximum principle. Similar assertions are valid with “min” replacing “max”. Interpretation. So if u attains its maximum (or minimum) at an interior point, then u is constant at all earlier times. This accords with our strong intuitive understanding of the variable t as denoting time: the solution will
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