We give the radial solution of the heat equation \(PU(x,t)=F(x,t,U(x,t))\) where \(x=(x_ 1,x_ 2,x_ 3)\), \(P=\Delta_ x-D_ t+C(t)\), \(\Delta_ x\) is the Laplace operator in the domain \[ D=D_ 1\times T,\quad D_ 1=\{x: 0<a<| x| <b\},\quad T=\{t: 0<t<t_ 1\}, \] satisfying the initial condition \[ U(x,0)+Q(U(x,0))=F_ 1(x)\quad for\quad x\in D_ 1 \] and the boundary value conditions \[ U(x,t)+K_ 1(U(x,t))=F_ 2(x,t)\quad for\quad (x,t)\in D_ 2\times T,\quad D_ 2=\{x: | x| =a\}, \]
\[ U(x,t)+K_ 2(U(x,t))=F_ 3(x,t)\quad for\quad (x,t)\in D_ 3\times T,\quad D_ 3=\{x: | x| =b\}, \] where \(F,F_ i\) \((i=1,2,3)\), \(Q,K_ i\) \((i=1,2)\) are the given functions.