How to set the variable with Fourier Transform when solving PDEs using coefficient form?

Hello everyone
My question can be simplified below:
AIM: get a dependent field u(x,t)
PDE general form: u_t=A(u,u_x)+B(u)
Want to solve it using coefficient form.

Problem:
A(u,u_x) can be expressed explicitly. However, the term B(u) is given by a Fourier Transform. I should first transform u to Fourier Space U, then calculate an integration related U to get B(u).
I know the Built-in Function Integrate cannot be used in user-defined function.

How I can add B(u) expressed by inverse Fourier Transform into the coefficient form?

Thanks
Hoping to hear from everyone.

Jingtian