Suppose U := (0, 1) CR and a smooth enough function u : U x R,...

 

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Suppose U := (0, 1) CR and a smooth enough function u : U x R, R satisfies x€ U, t> 0 u(0, t) = g1(t), u(1, t) = g2(1) t> 0 x€ U x € U, Un - Uxx = f(x, 1) u(x, 0) = 0(x) u,(x, 0) = (x) for f e C(U x R.), 4,y e C(U) and g1, 92 E C(R,). Prove, using the energy method, that there is at most one solution to the above system. (Hint: Suppose there are two distinct solutions u and v and take the inner product of the PDE for w:= u- v with w, to reach a contradiction.) Suppose U := (0, 1) CR and a smooth enough function u : U x R, R satisfies x€ U, t> 0 u(0, t) = g1(t), u(1, t) = g2(1) t> 0 x€ U x € U, Un - Uxx = f(x, 1) u(x, 0) = 0(x) u,(x, 0) = (x) for f e C(U x R.), 4,y e C(U) and g1, 92 E C(R,). Prove, using the energy method, that there is at most one solution to the above system. (Hint: Suppose there are two distinct solutions u and v and take the inner product of the PDE for w:= u- v with w, to reach a contradiction.)

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we will use energy equation of the wave equation to prove that there is an atmost one ... View the full answer