Question

Let \( u(x, t) \) be the solution of the initial value problem \[ \frac{\partial u}{\partial t} + 3 \frac{\partial u}{\partial x} = u, \quad x \in \mathbb{R}, \quad t>0, \quad u(x, 0) = \cos x, \] and let \( v(x, t) \) be the solution of the initial value problem \[ \frac{\partial v}{\partial t} + 3 \frac{\partial v}{\partial x} = v^2, \quad x \in \mathbb{R}, \quad t>0, \quad v(x, 0) = \cos x. \] Then, which of the following is/are TRUE?

• A. \( |u(x,t)| \leq e^t { for all } x \in \mathbb{R} { and for all } t>0 \)
• B. \( v(x, 1) { is not defined for certain values of } x \in \mathbb{R} \)
• C. \( v(x, 1) { is not defined for any } x \in \mathbb{R} \)
• D. \( u(2\pi, \pi) = -e^\pi \)

Hint:

For first-order linear PDEs, solutions typically exhibit exponential growth, and for nonlinear PDEs, the solution may blow up depending on initial conditions. In this case, the solution for \( v(x, t) \) becomes undefined for certain values of \( x \).

Answer 1

The Correct Option is A, B, D

Step 1: Understanding the Problem
The given equations are both first-order linear PDEs for \( u(x, t) \) and \( v(x, t) \). For \( u(x, t) \), the solution involves standard methods for solving first-order linear PDEs, and the form of the solution satisfies \( |u(x,t)| \leq e^t \) as the exponential growth is bounded by \( e^t \).

Step 2: Analyzing \( v(x, t) \)
The equation for \( v(x, t) \) is nonlinear due to the \( v^2 \) term. This type of nonlinearity can cause the solution to become undefined for certain values of \( x \) and \( t \), as the solution may blow up.

Step 3: Evaluating \( u(2\pi, \pi) \)
Given the structure of the equation for \( u(x,t) \), evaluating \( u(2\pi, \pi) \) gives \( -e^\pi \), as it follows the solution pattern for this type of equation.

Step 4: Conclusion
Thus, the correct answers are \( \boxed{A} \), \( \boxed{B} \), and \( \boxed{D} \).

Final Answer
\[ \boxed{A} \quad |u(x,t)| \leq e^t \text{ for all } x \in \mathbb{R} \text{ and for all } t > 0 \]
\[ \boxed{B} \quad v(x, 1) \text{ is not defined for certain values of } x \in \mathbb{R} \]
\[ \boxed{D} \quad u(2\pi, \pi) = -e^\pi \]