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

To solve the given initial value problems (IVP) and determine the validity of the provided options, we will analyze the behavior of the solutions \( u(x, t) \) and \( v(x, t) \). Let's break down each scenario:

1. Problem for \( u(x, t) \): 

The given equation is:

1. \(\frac{\partial u}{\partial t} + 3 \frac{\partial u}{\partial x} = u, \quad u(x, 0) = \cos x\)

This is a first-order linear partial differential equation. Using the method of characteristics, the characteristic equations are:

1. \(\frac{dx}{dt} = 3, \quad \frac{du}{dt} = u\)

The solution to these characteristic equations gives:

1. \(x(t) = x_0 + 3t\)

2. \(u(t) = u_0 e^t\), where \(u_0 = \cos(x_0)\)

Substituting back, since \(x_0 = x - 3t\), we have:

1. \(u(x, t) = \cos(x - 3t) e^t\)

To evaluate \(u(2\pi, \pi)\):

1. \(u(2\pi, \pi) = \cos(2\pi - 3\pi) \cdot e^\pi = \cos(-\pi) \cdot e^\pi = -1 \cdot e^\pi = -e^\pi\)

Thus, the statement \(u(2\pi, \pi) = -e^\pi\) is true.

1. Boundedness of \( u(x, t) \):

We know:

1. \(|u(x, t)| = |\cos(x - 3t)| \cdot e^t\)

Since \(|\cos(x - 3t)| \leq 1\), it follows that:

1. \(|u(x, t)| \leq e^t\)

Thus, the statement \(|u(x, t)| \leq e^t \text{ for all } x \in \mathbb{R} \text{ and } t>0\) is true.

1. Problem for \( v(x, t) \):

The equation is:

1. \(\frac{\partial v}{\partial t} + 3 \frac{\partial v}{\partial x} = v^2, \quad v(x, 0) = \cos x\)

This is a Burgers-type equation and can exhibit shock formation or blow-up behavior. To solve for \(v(x, 1)\), note that the nonlinearity \(v^2\) can cause singularities.

Assuming a similar method of characteristics approach, we get:

1. \(\frac{dv}{dt} = v^2\), leading to a blow-up in finite tim\)

For certain \(x\), the value of \(v(x, t)\) may become undefined, due to finite-time blow-up. Specifically, if at \(t=1\), for certain values of \(x\), \(|v|\) can approach infinity.

Thus, the statement \( v(x, 1) \) is not defined for certain values of \( x \in \mathbb{R} \) is true.

Therefore, the valid statements are:

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