Fourier Transform

Let u(:c,t) solve the partial differential equation
c’3;u=8,2,u-u, a:E1R, t >0,
with initial condition u(a:,0) = v(a:) for (E E R. Furthermore, denote by 11 = J-‘ the Fourier transform of
u with respect to 9:. Show that 12 satisfies the equation
ata = -(k2 + 1)a

with initial condition 1l(k,O) = 0(k) = }'{v}. Find the solution 11 of the equation in Fourier space and
thereby the solution u of the original equation.

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