Suppose we have $u_t = a^2 u_{xx}$, $0 < x < 1 , t > 0$, boundary conditions are $u(0,t) = u(1,t) = 0$, and the initial condition is $u(x,0) = \sin(\pi x)$. What will be the behavior of $u(x,t)$ as time increases. There may be more than one correct answer. You do not need to solve the equation to answer this question.

A. For a fixed t, the second x derivative, $u_{xx}$, will always be negative.

B. For a fixed t, the solution will look kind of like an inverted parabola.

C. For a fixed t, the solution will be linear

D. The solution does not change as time goes on.

E. The solution behaves unpredictably.

F. The solution decreases slowly towards $u=0$.

G. The solution decreases to minus infinity.

H. For a fixed t, the second x derivative, $u_{xx}$, will take on both signs.

I. The solution increases to infinity.

J. The solution increases slowly towards $u=0$.

K. None of the above