partial-differential-equations-by-evans · Problem 10

Nonlinear First-Order PDE

Mathematics — Calculus · heat-equation-scaling

Problem

Suppose

u

is smooth and solves

u_t - \Delta u = 0

\mathbb{R}^n \times (0,\infty)

. (i) Show

u_\lambda(x,t) := u(\lambda x, \lambda^2 t)

also solves the heat equation for each

\lambda \in \mathbb{R}

. (ii) Use (i) to show

v(x,t) := x \cdot Du(x,t) + 2tu_t(x,t)

solves the heat equation as well.