The heat equation is the model parabolic equation. Its defining features are instantaneous smoothing, dissipation of spatial oscillation, positivity, and irreversibility. These properties appear simultaneously in its Gaussian, Fourier, energy, and semigroup representations.
The heat kernel on Euclidean space
Definition. For \(t>0\), the heat kernel on \(\mathbb R^n\) is \[ \Gamma_t(x)=\frac{1}{(4\pi t)^{n/2}} \exp\!\left(-\frac{|x|^2}{4t}\right). \] It satisfies \(\Gamma_t\ge0\), \(\int_{\mathbb R^n}\Gamma_t\,\mathrm{d}x=1\), and \(\Gamma_t*\Gamma_s=\Gamma_{t+s}\).
Theorem (Cauchy problem). If \(u_0\in L^p(\mathbb R^n)\), \(1\le p\le\infty\), then \[ u(x,t)=(\Gamma_t*u_0)(x) \] solves \(u_t-\Delta u=0\) for \(t>0\), and \(u(\cdot,t)\to u_0\) in \(L^p\) as \(t\downarrow0\) when \(p<\infty\). For every multi-index \(\alpha\), \[ \|\partial^\alpha u(\cdot,t)\|_p \le C_{\alpha,n}t^{-|\alpha|/2}\|u_0\|_p . \]
Proof. Direct differentiation gives \(\partial_t\Gamma_t=\Delta\Gamma_t\); differentiation under the convolution therefore proves the equation. The family \((\Gamma_t)\) is an approximate identity, which gives convergence to \(u_0\). Finally, \[ \partial^\alpha u=(\partial^\alpha\Gamma_t)*u_0, \qquad \|\partial^\alpha\Gamma_t\|_1 =t^{-|\alpha|/2}\|\partial^\alpha\Gamma_1\|_1, \] and Young's convolution inequality yields the estimate.
The estimate says more than existence: every positive time turns rough \(L^p\) data into a smooth function. The cost \(t^{-|\alpha|/2}\) records the parabolic scaling \(x\sim t^{1/2}\).
Fourier representation and dissipation
With \(\widehat u(\xi,t)=\int e^{-ix\cdot\xi}u(x,t)\,\mathrm{d}x\), the equation becomes \[ \partial_t\widehat u(\xi,t)+|\xi|^2\widehat u(\xi,t)=0, \qquad \widehat u(\xi,t)=e^{-t|\xi|^2}\widehat u_0(\xi). \] High frequencies are suppressed faster than low frequencies. This is the spectral origin of smoothing and the reason the backward heat equation is unstable.
Proposition (energy identity). A sufficiently regular solution of \(u_t-\Delta u=0\) on a bounded domain \(\Omega\), with \(u=0\) on \(\partial\Omega\), satisfies \[ \frac12\frac{\mathrm{d}}{\mathrm{d}t}\|u(t)\|_{L^2(\Omega)}^2 +\|\nabla u(t)\|_{L^2(\Omega)}^2=0. \] Consequently the solution with prescribed initial data is unique.
Proof. Multiply the equation by \(u\), integrate, and use the boundary condition: \[ \int_\Omega uu_t\,\mathrm{d}x -\int_\Omega u\Delta u\,\mathrm{d}x =\frac12\frac{\mathrm{d}}{\mathrm{d}t}\|u\|_2^2+\|\nabla u\|_2^2=0. \] Apply this identity to the difference of two solutions. Zero initial energy forces the difference to vanish.
Positivity, mass, and comparison
On \(\mathbb R^n\), positivity of \(\Gamma_t\) gives \[ u_0\ge0\Longrightarrow u(\cdot,t)\ge0,\qquad \int u(x,t)\,\mathrm{d}x=\int u_0(x)\,\mathrm{d}x. \] The parabolic maximum principle gives the corresponding comparison statement on domains: if \(u_0\le v_0\) and the lateral boundary data are ordered, then \(u\le v\). Thus heat flow is order preserving as well as \(L^2\)-dissipative.
Bounded domains and forcing
Theorem (spectral solution). Let \((\lambda_k,\phi_k)\) be the Dirichlet eigenpairs of \(-\Delta\) on a smooth bounded domain, orthonormal in \(L^2(\Omega)\). For \(u_0=\sum_k u_k\phi_k\), \[ u(t)=\sum_{k=1}^{\infty}e^{-\lambda_k t}u_k\phi_k \] is the unique \(L^2\)-solution. For \(u_t-\Delta u=f\), the variation-of-constants formula is \[ u(t)=e^{t\Delta}u_0+\int_0^t e^{(t-s)\Delta}f(s)\,\mathrm{d}s. \]
Example (loss of information backward in time). On \((0,\pi)\), initial data \(u_0^{(k)}(x)=k^{-1}\sin(kx)\) tend to zero in \(L^2\), while at a fixed terminal time \(T\), recovering them from \(u^{(k)}(T)=k^{-1}e^{-k^2T}\sin(kx)\) requires multiplication by \(e^{k^2T}\). Arbitrarily small high-frequency observation errors are therefore amplified without bound. The forward problem is well posed; the backward problem is not.