# A question on a solution of an inhomogeneous heat equation.

I am now working on the following PDE equation (Evan's PDE textbook Section 2.5 No.14) \begin{align} u_{t}-\Delta u + cu=f \ \ & on \ \ \mathbb{R}^n\times (0,\infty) \\ u=g \ \ & on \ \ \mathbb{R}^n\times \{0\}, \end{align} where $f$ and $g$ are smooth functions on the domains above. Then the Fourier transform with respect to spacial variables yields \begin{align} \hat{u}_{t}+|\xi|^2\hat{u} + c\hat{u}=\hat{f} \ \ & on \ \ \hat{\mathbb{R}}^n\times (0,\infty) \\ \hat{u}=\hat{g} \ \ & on \ \ \hat{\mathbb{R}}^n\times \{0\}. \end{align} Then this ODE can be solved as $$\hat{u}_{t}=e^{-t(c+|\xi|^2)}\hat{g}+\int_{0}^{t}e^{(s-t)(c+|\xi|^2)}\hat{f}ds,$$ and therefore we get $$u(x,t)=e^{-ct}(\phi_{H}*g)(x,t)+\int_{0}^{t}e^{(s-t)(c+|\xi|^2)}\hat{f}ds.$$ I wonder if this is what the author expect readers to derive. This seems quite complicated. Can it be reduced to be an simpler form than this?

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He might be joking usually, but in this case there is a nicer way. He would instead advise the following: Consider the change of variables $v(x,t) = e^{ct} u(x,t)$. I think you can take it from there.