Diffusion Equation Homework: Show Formula (8) Makes Sense for 0 < t < 1/(4ak)

[Dassinia]

Homework Statement

Hello, I don't understand the solution of an exercise

Let P(x) be a continuous function such that |P(x)|≤Ce^ax² .
Show that formula (8) for the solution of the diffusion equation makes sense for 0 < t < 1/(4ak), but not necessarily for larger t.

Homework Equations

Equation (8) refers to
u(x,t)=1/√(4*pi*k*t) ∫ e^-(x-y)²/4ktP(y) dy

The Attempt at a Solution

The solution just says that

I don't understand how they conclude ? I don't see why we have 0 < t < 1/(4ak)

Thanks

 

[vela]

Consider the sign of the $y^2$ term in the exponent. What has to be true for the integral to converge?

 

[Dassinia]

Consider the sign of the $y^2$ term in the exponent. What has to be true for the integral to converge?

I see, thanks !