7 2 J File Preview Problem 5 (15 points): Fix a positive real number L. Suppose that u R× [0,T] → R is a positive solution of the heat equation Ut(x,t) = Uxx(x,t) which is 2L-periodic in the first variable. In other words, u(x+2L,t) = u(x,t) for all x and t. We define and = L J(t) J-L u(x, t) dx S(t) = − √ u(x, t) log u(x, t) dr. -L (i) Please show that the derivatives of J(t) and S(t) are given by J'(t) = 0 2 and S'(t) = [[ u(x,t) =¹ u₂(x, t)² dx -L for 0≤t≤T. (Hint: Use integration by parts.) (ii) Please show that the second derivative of S(t) is given by L S" (t) = -2 - 2 [ ", u(x, t)~^¹ [uzz(x, t) — u(x, t)¯¹ uz(x,t)²]² dx - -L for 0<t<T. (Hint: Use (i) and integration by parts.) (iii) We next define - log t 2 W(t) = t S' (t) + S(t) − J(t). Please show that the derivative of W(t) can be written in the form W'(t) = −2t L 1 2 dx ·L u(x,t)¯¹ [uxx(x,t)—u(x,t)¯¹uz(x, t)² + 1 = u(x, t)]². -L 2t for 0 <t<T. Deduce from this that the function W(t) is monotone decreasing for 0 <t<T.