First Order Equations Consider the following Cauchy problem with initial data So

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First Order Equations Consider the following Cauchy problem with initial data Solve the problem for u(x, t). Assume the problem is the result of the conservation of mass in a tube with pure advection, without any source term, and with time varying cross-sectional area A(t) > 0, and velocity, v(x, t) = xe-1. What equation does A(t) satisfy? What is the general function A(t)? Show the previous problem is well-posed proving the three conditions of well-posedness. (Hint: If there are two solutions, u1 and u2, what does w = u1 - u2 solve?)

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