1.* Suppose U := (0, 1) C R and a smooth enough function u : U x R, → R satisfies χε U t> 0 Un – Uxx = f(x, t) u(0, 1) %3D 9:(), и(1,1) %3 92(1) 1>0 u(х,0) %3 Ф(х) u,(x, 0) = µ(x) χεU x € U, for f e...

Extracted text: 1.* Suppose U := (0, 1) C R and a smooth enough function u : U x R, → R satisfies χε U t> 0 Un – Uxx = f(x, t) u(0, 1) %3D 9:(), и(1,1) %3 92(1) 1>0 u(х,0) %3 Ф(х) u,(x, 0) = µ(x) χεU x € U, for f e C(U × R,), ø, Y e C²(U) and g1, 92 € 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.)