(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose that u(x,t) satisfies the heat equation u_{t}=u_{x x} for 0<x<L and t>0 with initial condition u(x,0)=θ(x) and boundary conditions u(0,t)=u(L,t)=0. Suppose that θ(x)>0 for 0<x<L. Explain why u(x,t)>0 for all 0<x<L and t>0

2. Relevant equations

Strong Maximum principle: For nonconstant solutions to the heat equation, the maximum value of u(x,t) in the rectangle 0≤x≤L, 0≤t≤∞ can only occur for either x=0, x=L or t=0

Note: u_{x x} means doble partial derivative with respect x

3. The attempt at a solution

The maximum principle is telling us that the max value of u(x,t) should be in θ(x). Let's call the maxium value u_{max}. In this point the concavity should be negative, thus umax _{x x} < 0, and from the heat equation umax_{t} <0 telling us that the value of u will decrease in time ... but I just don't see why this can't go lower than zero... any help?

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Heat equation and maximum principle

**Physics Forums | Science Articles, Homework Help, Discussion**