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Given the initial-boundary value problem $$ u_t −u_{xx} = 2, \ \ \ \ \ \ x ∈ [−1,1], t ≥ 0,$$ With initial and boundary conditions $$u(x,0) = 0$$ $$ u(−1, t) = u(1, t)=0$$ Claim: the solution, $u(x,t)$, is such that $$ u(x, t) ≤ −x^2 + 1$$ for all $x ∈ [−1,1], \ \ t ≥ 0$. My idea: this is of course the heat equation and I am asked to prove that the solution cannot exceed a certain function, namely $-x^2+1$(which solves the heat equation), since the solution must "diffuse" in time. I was thinking that minimum principle should be involved here but I am not quite sure how to show the claim. Thanks for the help.

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The solution consists of two parts: the stationary solution $-x^2+1$ plus transient part, which is simply the solution to the homogeneous heat equation with the same boundary conditions. This solution can be easily found in explicit form (note that the initial condition will change). Do this and show that the transient solution is always less than 0.

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  • $\begingroup$ Thank you, that helps a lot! Just one thing: how will the initial condition change? $\endgroup$
    – johnsteck
    Oct 19, 2013 at 19:04
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    $\begingroup$ Since this is homework, here is another hint: put $u=-x^2+1+v$ and find equation, boundary and initial conditions for $v$. $\endgroup$
    – Artem
    Oct 19, 2013 at 19:09
  • $\begingroup$ Ok, I got it, thanks! $\endgroup$
    – johnsteck
    Oct 19, 2013 at 20:27

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