Conditional stability in determination of force terms of heat equations in a rectangle

We consider a heat equation in a rectangle: @[email protected]?t(x"1,"2,t)[email protected](x"1,x"2,t)[email protected](t)f(x"1,x"2), (x"1x"2)@?(0,1)x(0,1),0 0 and depends only on time t, we prove: (1) f(x"1,x"2) (0 < x"1,x"2 < 1) can be uniquely determined from the base boundary data u(x"1,0,t) (0 < x"1 < 1, 0 < t < T). (2) If f is restricted to a compact set in the Sobolev spaces, then we get an estimate:@[email protected]?"L"^"[email protected]^-^@bas @[email protected]?u(.,0,[email protected]?)"^"1"("("0","T")";"L"^"2"("0","1")")@70. Here the exponent @b is given by the order of the Sobolev space which is assumed to contain the set of f's.