6.2: Inhomogeneous Heat Equation
( \newcommand{\kernel}{\mathrm{null}\,}\)
Here we consider the initial value problem for u=u(x,t), u∈C∞(Rn×R+),
ut−△u=f(x,t) in x∈Rn, t≥0,u(x,0)=ϕ(x),
where ϕ and f are given. From
$$ \widehat{u_t-\triangle u}=\widehat{f(x,t)} \]
we obtain an initial value problem for an ordinary differential equation:
dˆudt+|ξ|2ˆu=ˆf(ξ,t)ˆu(ξ,0)=ˆϕ(ξ).
The solution is given by
$$\widehat{u}(\xi,t)=e^{-|\xi|^2 t}\widehat{\phi}(\xi)+\int_0^t\ e^{-|\xi|^2(t-\tau)}\widehat{f}(\xi,\tau)\ d\tau.\]
Applying the inverse Fourier transform and a calculation as in the proof of Theorem 5.1, step (vi), we get}
u(x,t)=(2π)−n/2∫Rn eix⋅ξ(e−|ξ|2tˆϕ(ξ) +∫t0 e−|ξ|2(t−τ)ˆf(ξ,τ) dτ) dξ.
From the above calculation for the homogeneous problem and calculation as in the proof of Theorem 5.1, step (vi), we obtain the formula
u(x,t)=1(2√πt)n∫Rn ϕ(y)e−|y−x|2/(4t) dy+∫t0∫Rn f(y,τ)1(2√π(t−τ))n e−|y−x|2/(4(t−τ)) dy dτ.
This function u(x,t) is a solution of the above inhomogeneous initial value problem provided
$$\phi\in C(\mathbb{R}^n),\ \ \sup_{\mathbb{R}^n}|\phi(x)|<\infty\]
and if
$$f\in C(\mathbb{R}^n\times[0,\infty)),\ \ M(\tau):=\sup_{\mathbb{R}^n}|f(y,\tau)|<\infty,\ 0\le\tau<\infty.\]
Contributors and Attributions
Integrated by Justin Marshall.