Question

Let \( u(x, t) \) be the solution of the initial-boundary value problem \[ \frac{\partial u}{\partial t} = 2 \frac{\partial^2 u}{\partial x^2}, \quad 0<x<1, \quad t>0, \] with the boundary conditions \[ u(0, t) = u(1, t) = 0, \quad u(x, 0) = 2x(1 - x). \] Then, which of the following is/are TRUE?

• A. \( 0 \leq u(x, t) \leq \frac{1}{4} { for all } t \geq 0 { and } x \in [0, 1] \)
• B. \( u(x, t) = u(1 - x, t) { for all } t \geq 0 { and } x \in [0, 1] \)
• C. \( \int_0^1 (u(x, t))^2 dx \) is a decreasing function of \( t \)
• D. \( \int_0^1 (u(x, t))^2 dx \) is not a decreasing function of \( t \)

Hint:

For heat equations with symmetric initial conditions, the solution remains symmetric. Additionally, the total energy (integral of \( u(x,t)^2 \)) decreases over time as heat dissipates.

Answer 1

The Correct Option is B, C

Step 1: Understanding the Problem
This is a classic heat equation with initial and boundary conditions. The solution will be symmetric about \( x = \frac{1}{2} \), so \( u(x, t) = u(1 - x, t) \) for all \( t \geq 0 \).

Step 2: Evaluating \( u(x, t) \)
The initial condition \( u(x, 0) = 2x(1 - x) \) is symmetric, and the solution will remain symmetric for all times. The integral \( \int_0^1 (u(x, t))^2 dx \) represents the total energy, and it will decrease over time as the heat dissipates, thus making it a decreasing function of \( t \).

Step 3: Conclusion
Thus, the correct answers are \( \boxed{B} \) and \( \boxed{C} \).

Final Answer
\[ \boxed{B} \quad u(x, t) = u(1 - x, t) \text{ for all } t \geq 0 \text{ and } x \in [0, 1] \]
\[ \boxed{C} \quad \int_0^1 (u(x, t))^2 dx \text{ is a decreasing function of } t \]