# Understanding the proof for the comparison principle for the heat equation.

I am studying the principle of comparison for heat equations and I am having some problems understanding the proof.

The theorem is:

Let $$\Omega\subset\mathbb{R}^n$$ be open and bounded, $$0 and let $$u,v\in{C}(\overline\Omega\times[0,T))$$ solve $$u_t-\Delta{u}=0$$ and $$v_t-\Delta{v}=0$$ respectively in $$\Omega_T:=\Omega\times(0,T)$$. Then if $$u\leq{v}$$ on $$\partial\Omega\times(0,T)\cup\Omega\times\{t=0\}=:\partial_p\Omega_T$$, $$u\leq{v}$$ on $$\overline\Omega\times[0,T]$$.

The Proof first defines $$w=u-v$$, $$w^+=(u-v)^+=max\{0, u-v\}$$ and considers the integral of the product of $$w^+$$ and $$w_t-\Delta{w}=0$$ on $$\Omega_S$$, for $$0. It is stated that $$w_tw^+=\frac{\partial}{\partial{t}}\cdot\frac{1}{2}(w^+)^2$$ and that $$\nabla{w}\nabla{w^+}=|\nabla(w^+)|^2$$, nearly everywhere. Here I don't understand how we get this results.

Then the proof moves on to the integral 0=$$\int_{0}^{S}\int_{\Omega}(w_tw^+-\Delta{w}w^+)=\int_{0}^{S}\int_{\Omega}\frac{\partial}{\partial{t}}\frac{(w^+)^2}{2}+|\nabla(w^+)^2|dxdt-\int_{0}^{S}\int_{\partial\Omega}\frac{\partial{w}}{\partial{n}}w^+dSdt$$, since I didn't understand the step before, I also don't understand this one now. I am also not quite sure how to get $$\int_{0}^{S}\int_{\partial\Omega}\frac{\partial{w}}{\partial{n}}w^+dSdt$$.

Lastly, $$\int_{0}^{S}\int_{\Omega}\frac{\partial}{\partial{t}}\frac{(w^+)^2}{2}+|\nabla(w^+)^2|dxdt-\int_{0}^{S}\int_{\partial\Omega}\frac{\partial{w}}{\partial{n}}w^+dSdt\geq\frac{1}{2}\int_{\Omega}(w^+(S))^2dx-\frac{1}{2}\int_{\Omega}(w^+(0=\frac{1}{2}\int_{\Omega}(w^+(S))^2dx))^2dx$$, this follows because $$w^+=0$$ on $$\partial\Omega\times(0,T)$$ and also on $$\Omega\times\{t=0\}$$. This means that $$w^+=0$$ on $$\Omega\times\{t=S\}$$, for all $$0.Therefore $$u\leq{v}$$ on $$\Omega_T$$.

This last step I understand, my problem is the first two. Any help would be greatly appreciated.

I agree that this "proof " looks fishy. A simpler alternative classical proof is outlined below.

Definition. A strict sub-solution of the heat equation on the closed "tin-can" $$R=\overline{\Omega}\times[0,T]$$ is a smooth function $$u$$ that satisfies $$u_t-\triangle u<0$$ on the interior of $$R$$.

That is, $$u$$ satisfies the inhomogeneous equation $$u_t= \triangle u -f$$ where $$f>0$$. In physical terms, $$f$$ represents the internal rate in $$R$$ at which heat is being withdrawn from the system. Intuitively we expect this energy loss $$f$$ forces the system to "wind down" as time progresses. Here is a precise statement that is in accord with that intuition.

Lemma (Weak Maximum Principle). A strict sub-solution has no local maximum in the interior of $$R$$.

Proof of lemma. At any hypothetical interior max, the critical point condition would force $$u_t=0$$, which would force $$\triangle u>0$$, which violates the Second Derivative Test for a local max (stating that at a local max no second-order directional derivative can be positive.)

The lemma can easily be strengthened to state that the maximum cannot occur on the interior of the top face $$\Omega \times \{T\}$$. (Essentially the same proof works, since in this case at the endpoint time, the occurrence of a maximum requires $$u_t\geq 0$$.)

Proof of Comparison Principle.

To prove the Comparison Principle, you want to show that the function $$w= u-v$$ that is negative on the bottom and sides of the can stays negative in the interior of the can and on the top lid. If we knew that $$w$$ were actually a strict sub-solution of the heat equation, the lemma would give us the desired conclusion immediately.

The final technical detail in the proof is showing that you can perturb our given $$w$$ arbitrarily slightly to make it a strict sub-solution that is still negative on the bounding bottom and sides. That can be done by constructing practically any convenient explicit perturbing sub-solution that is small on these bounding edges, and adding it to $$w$$.

Example: Perturb $$w$$ by adding a very small multiple of $$h=||x||^2 e^{-Mt}$$. Note that $$h$$ has the property that $$h_t -\triangle h= -Mh - Ce^{-Mt}$$ is negative, and any sufficiently small multiple of it will not change the boundary values of $$w$$ significantly.

P.S.The ideas involved in the proof of the Maximum Principle are useful because they generalize easily to apply to many nonlinear equations too, such as $$u_t - \sqrt{1+u_x^2 + u_y^2} (\triangle u)=0$$.

• Thank you, this prove is easier to understand. The one I posted is from my lecturers notes. Apr 11, 2023 at 16:03