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a) Find the solution with $u(x,0)=f(x).$

b) If $f(x)=\left\{\begin{array}{cl}\cos^2x,&\text{if }-\frac\pi2\le x\le\frac\pi2,\\

0,&\text{in the rest}.\end{array}\right.$

Describe $u(x,t)$ for $t\ge0.$

I have to use Fourier transform, but don't know how to apply it for $u_x\cos t.$ As for part b), I don't know how to describe $u(x,t).$

Thanks for the help!