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Homework Help: Heat equation and maximum principle

  1. Jun 20, 2012 #1
    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?
  2. jcsd
  3. Jun 20, 2012 #2


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    Homework Helper

    If there's a "Strong Maximum Principle" shouldn't there also be a "Strong Minimum Principle"? Think about negating your boundary conditions and changing u(x,t) to -u(x,t).
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