For what values of the velocity $c$ does the equation $u_t = u -u^2 + u_{xx}$ have a solution in the form of a traveling wave $u = \varphi(x-ct)$, $\varphi(-\infty) = 1$, $\varphi(\infty) = 0$, $0 \le u \le 1$?
To determine the values of the velocity \( c \) for which the equation \( u_t = u - u^2 + u_{xx} \) has a solution in the form of a traveling wave \( u = \varphi(x-ct) \), we can follow these steps:

1. **Substitute the traveling wave form into the PDE:**
   Let \( \xi = x - ct \). Then \( u = \varphi(\xi) \). We have:
   \[
   u_t = \frac{\partial \varphi}{\partial t} = -c \varphi'(\xi)
   \]
   and
   \[
   u_{xx} = \frac{\partial^2 \varphi}{\partial x^2} = \varphi''(\xi)
   \]
   Substituting these into the original PDE \( u_t = u - u^2 + u_{xx} \), we get:
   \[
   -c \varphi'(\xi) = \varphi(\xi) - \varphi^2(\xi) + \varphi''(\xi)
   \]

2. **Rewrite the equation in terms of \(\varphi\):**
   \[
   \varphi''(\xi) - c \varphi'(\xi) + \varphi(\xi) - \varphi^2(\xi) = 0
   \]

3. **Analyze the resulting ODE:**
   This is a second-order nonlinear ordinary differential equation. To find the conditions on \( c \) for which there exists a traveling wave solution, we refer to known results for similar equations.

4. **Reference to Fisher-Kolmogorov equation:**
   The given equation is a form of the Fisher-Kolmogorov equation, which is known to admit traveling wave solutions. According to the theory of traveling waves for the Fisher-Kolmogorov equation, the wave speed \( c \) must satisfy:
   \[
   c \geq 2
   \]
   This result can be found in various references, such as the Wikipedia page on Fisher's equation and the MathWorld page on Fisher's equation.

5. **Conclusion:**
   Therefore, the values of \( c \) for which the equation \( u_t = u - u^2 + u_{xx} \) has a traveling wave solution \( u = \varphi(x-ct) \) are those for which \( c \geq 2 \).

The final answer is \( \boxed{ c \geq 2 } \)