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Maximum Principle

Let v = u + e x^2, e>0.
dv/dt = du/dt
d^2v/dx^2 = d^2u/dx^2 + e
we have L(v)=-ek<0.
Let M=max{ v(x,t) : (x,t) in R }.
Claim1. v can not attain max in the int(R), the interior of R.
  Let v(x0,t0) = M for some (x0,t0) in int(R).
  => dv/dt(x0,t0)=0 and d^2v/dx^2(x0,t0) leq 0.
  => L(v(x0,t0))=0-k d^2v/dx^2(x0,t0) geq 0. -><-
Claim2. v can not attain max at t = T.
  Let v(x0,T)=M for some x0 in (0,l).
  => v(x0,T) geq v(x0,T-h), 0 < h < T.
  => dv/dt(x0,T) = lim_{h->0} ( v(x0,T) - v(x0,T-h) ) / h geq 0.
  => L(v(x0,T)) geq 0. -><-.
=> max(v) is assumed either initially or on the lateral sides for all e>0.
=> max(u) is assumed either initially or on the lateral sides.

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