3. (a) Give the precise statement of the weak and strong maximum principle for a solution...

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3. (a) Give the precise statement of the weak and strong maximum principle for a solution u of ut = kur,a on [0, 1] × [0, T] for T > 0. (2 points) (b) Consider the diffusion equation u = kura on (0, 1) x (0, T) with initial value u(x, 0) u(1, t) = 0. Show that 0 < u(x, t) < 1 for all t e (0, T| and x E (0, 1). (You may use part (a) if you like.) (4 points) 4.x(1 x) on [0, 1] and boundary condition u(0, t) (c) Show that u(x, t) = u(1 – x,t) for all t e (0, T] and x E [0, 1]. (4 points) 3. (a) Give the precise statement of the weak and strong maximum principle for a solution u of ut = kur,a on [0, 1] × [0, T] for T > 0. (2 points) (b) Consider the diffusion equation u = kura on (0, 1) x (0, T) with initial value u(x, 0) u(1, t) = 0. Show that 0 < u(x, t) < 1 for all t e (0, T| and x E (0, 1). (You may use part (a) if you like.) (4 points) 4.x(1 x) on [0, 1] and boundary condition u(0, t) (c) Show that u(x, t) = u(1 – x,t) for all t e (0, T] and x E [0, 1]. (4 points)

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