= Weak parabolic maximum principle
For a classical solution of $u_t-\Delta u+cu=0$ with $c\geq0$, the positive part of its maximum is bounded by the corresponding maximum on the initial and lateral <parabolic boundary>. When $c>0$, a positive maximum in space at an interior or final time gives $u_t\geq0$, $\Delta u\leq0$, and $u_t-\Delta u+cu>0$, a contradiction. For $c=0$, a small strictly increasing exponential time perturbation gives the weak inequality by a limiting argument. Applied to both signs of the difference of two solutions, the principle proves uniqueness for fixed initial and boundary data.
Back to article page
