Similarity solution

Remarks

Taking $\lambda = 1/t$ shows that $u$ is self-similar exactly when $$u(x, t) = t^{-\alpha}\, v\!\left( \frac{x}{t^{\beta}} \right)$$ for a function $v$ on $\R^n$, namely $v = u(\cdot, 1)$. Substituting this form turns the equation into an equation for $v$ in the variable $\eta = x / t^\beta$.

Example: the heat equation $u_t = \Delta u$ is invariant for $\beta = 1/2$ and every $\alpha$. The heat kernel $\Phi(x, t) = (4\pi t)^{-n/2} e^{-\abs{x}^2 / (4t)}$ is self-similar with $\alpha = n/2$, $\beta = 1/2$. Check: $\lambda^{n/2} \Phi(\lambda^{1/2} x, \lambda t) = \lambda^{n/2} (4\pi\lambda t)^{-n/2} e^{-\lambda \abs{x}^2/(4\lambda t)} = \Phi(x, t)$.

Non-example: $u(x, t) = e^{-t} \sin x$ solves $u_t = u_{xx}$ for $n = 1$, but it is not self-similar. A self-similar solution would have zero set $\sqrt{t}\, Z$ at time $t$, with $Z$ the zero set of $v$. But the zero set of $u(\cdot, t)$ is $\pi \Z$ for every $t$. At $t = 1$ this forces $Z = \pi \Z$, and then at $t = 4$ the zero set would be $2 \pi \Z \ne \pi \Z$.

For a scalar conservation law $u_t + F(u)_x = 0$, the scalings with $\alpha = 0$, $\beta = 1$ are symmetries. The corresponding similarity solutions $u = v(x/t)$ include the rarefaction waves and the solutions of the Riemann problem.