partial-differential-equations-by-evans · Problem 14

Nonlinear First-Order PDE

Mathematics — Calculus · heat-equation-subsolutions

Problem

We say

v \in C^2_1(U_T)

is a subsolution of the heat equation if

v_t - \Delta v \leq 0 \quad \text{in } U_T.

(a) Prove for a subsolution

v

that

v(x,t) \leq \frac{1}{4r^n} \iint_{E(x,t;r)} v(y,s) \frac{|x-y|^2}{(t-s)^2}\, dy\, ds

for all

E(x,t;r) \subset U_T

. (b) Prove that therefore

\max_{\bar{U}_t} v = \max_{\Gamma_T} v

. (c) Let

\phi : \mathbb{R} \to \mathbb{R}

be smooth and convex. Assume

u

solves the heat equation and

v := \phi(u)

. Prove

v

is a subsolution. (d) Prove

v := |Du|^2 + u_t^2

is a subsolution, whenever

u

solves the heat equation.