Section 6.3 Semilinear comparison principles
Having established maximum principles for linear parabolic operators, we now turn to semilinear operators of the form
\begin{gather}
\mathcal S(u) = -\partial_t u + a_{ij}\partial_{ij} u + F(x,t,u,\nabla u),\tag{6.4}
\end{gather}
where as always \(\nabla u = (\partial_1u,\ldots,\partial_Nu)\) refers to the gradient of \(u\) with respect to the spatial variable \(x\) alone. We write \(\mathcal S(u)\) and not \(\mathcal Su\) to emphasise the fact that \(\mathcal S\) is not a linear operator. Here as in Section 6.2, \(u\) is a \(C^2\) function on \(D = \Omega \times (0,T]\text{,}\) \(a_{ij}=a_{ji}\) are bounded functions on \(\overline D\text{,}\) and \(\mathcal S\) is uniformly parabolic if (6.2) holds for some constant \(\lambda_0 \gt 0\text{.}\) We assume that the nonlinear term in (6.4),
1
In previous version of this course (whose exams you have access to), \(\mathcal S(u)\) was simply written as \(Lu\text{.}\)
\begin{gather}
F=F(x,t,z,p), \qquad x \in \overline \Omega,\, t \in [0,T],\, p \in \R^N,\tag{6.5}
\end{gather}
is \(C^1\) jointly in all of its arguments, and we use the convention in (6.5) when taking partial derivatives of \(F\text{.}\) For example, we write \(F_{p_1} = \partial F/\partial p_1\) rather than the potentially confusing \(\partial_{\partial_1 u} F\) or \(F_{\partial_1 u}\text{.}\) Recall that the parabolic boundary is defined as \(\Sigma = (\partial\Omega \times (0,T]) \cup (\Omega \times \{0\})\text{.}\)
Theorem 6.7. Comparison principle.
Under the above assumptions, suppose that \(\Omega\) is bounded and \(\mathcal
S\) is uniformly parabolic. If \(u,v \in C^2(D) \cap C^1(\overline D)\) satisfy \(\mathcal S(v) \le \mathcal S(u)\) in \(D\) and \(v \ge u\) on \(\Sigma\text{,}\) then \(v \ge u\) in \(D\text{.}\)
Proof.
Consider the difference \(w=u-v\text{,}\) which by assumption satisfies \(w \le
0\) on \(\Sigma\) and
\begin{gather}
0 \le \mathcal S(u) - \mathcal S(v) = -\partial_t w + a_{ij} \partial_{ij} w
+ F(x,t,u,\nabla u) - F(x,t,v,\nabla v)\tag{6.6}
\end{gather}
in \(D\text{.}\) As in Section 5.3, the trick is to rewrite (6.6) as a linear differential inequality for \(w\text{.}\) For the sake of variety, we will use the mean value theorem this time rather than the fundamental theorem of calculus. To this end, we define a \(C^1\) function
\begin{gather*}
G(x,t,s) = F\big(x,t,su+(1-s)v,s\nabla u + (1-s)\nabla v\big),
\end{gather*}
so that the nonlinear term in (6.6) is \(G(x,t,1)-G(x,t,0)\text{.}\) Applying the mean value theorem to \(G\) as a function of \(s\text{,}\) there exists \(\theta = \theta(x,t) \in [0,1]\) such that
\begin{gather*}
G(x,t,1) -G(x,t,0) = G_s(x,t,\theta(x,t)).
\end{gather*}
Substituting the definition of \(F\) and using the chain rule, this becomes
\begin{align*}
\amp F(x,t,u,\nabla u) - F(x,t,v,\nabla v)\\
\amp\qquad= \underbrace{F_z\big(x,t,\theta u+(1-\theta)v,\theta\nabla u + (1-\theta)\nabla v\big)}_{=:\,c(x,t)}(u-v)\\
\amp\qquad\qquad + \underbrace{F_{p_i}\big(x,t,\theta u+(1-\theta)v,\theta\nabla u + (1-\theta)\nabla v\big)}_{=:\,b_i(x,t)}\big(\partial_i u - \partial_i
v\big)\\
\amp\qquad =: b_i \partial_i w + c w,
\end{align*}
where the summation convention is in force and we have suppressed the dependence of \(\theta\) on \((x,t)\text{.}\) Since \(F\) is \(C^1\) and \(\theta,u,v,\nabla u,\nabla v\) are all bounded, the coefficients \(b_i\) and \(c\) are also bounded. Plugging back into (6.6) we find that \(w\) satisfies
\begin{gather}
Lw := -\partial_t w + a_{ij}\partial_{ij} w + b_i \partial_i w + cw \ge 0.\tag{6.7}
\end{gather}
With (6.7) established, we can now simply apply the linear comparison principle Corollary 6.4 to \(w\) and the constant function \(0\text{.}\) By assumption we have \(w \le 0\) on \(\Sigma\text{,}\) and we have just shown that \(Lw \ge 0 = L0\) in \(D\text{.}\) Thus by Corollary 6.4 we have \(w \le 0\) in \(D\text{,}\) and hence \(u \le v\) in \(D\) as desired.
Note that, when the nonlinear function \(F(x,t,z,p)\) does not depend on \(p\text{,}\) the requirement \(u,v \in C^1(\overline D)\) in Theorem 6.7 can be weakened to \(u,v \in C^0(\overline D)\text{.}\)
Remark 6.8.
As in Section 5.3 for elliptic equations, it is comparison principles rather than maximum principles which generalise most readily to nonlinear problems.
Remark 6.9.
This strategy of reinterpreting nonlinear equations or inequalities (like (6.6)) as linear ones (like (6.7)) but with messy coefficients shows up all over the place in PDEs, not just for elliptic and parabolic partial differential equations and not just for maximum principle arguments. It is one of the (many!) motivations for developing linear theories which place only mild assumptions on the coefficients, and certainly for moving beyond the techniques for constant-coefficient problems which you may have seen in other units.
Such reinterpretations are more of an art than a science, and lend themselves naturally to problem sheet and exam questions.
Exercises Exercises
1. (PS10) Uniqueness for semilinear problems.
Let \(\Omega\) be bounded, \(f \in C^1(\R)\text{,}\) \(u_0 \in C^0(\overline \Omega)\) with \(u_0 = 0\) on \(\partial \Omega\text{,}\) and \(T \gt 0\text{.}\) Prove that the problem
\begin{align*}
\left\{
\begin{alignedat}{2}
-\partial_t u + \Delta u + f(u) \amp= 0 \amp\qquad\amp \ina D = \Omega \times (0,T],\\
u \amp= 0 \amp\qquad\amp \ona \partial\Omega \times (0,T],\\
u \amp= u_0 \amp\qquad\amp \ona \Omega \times \{0\},\\
\end{alignedat}
\right.
\end{align*}
has at most one solution \(u \in C^2(D) \cap C^1(\overline D)\text{.}\)
Hint.
Use the (semilinear) comparison principle.
Solution.
Let \(\mathcal S(u) = -\partial_t u + \Delta u + f(u)\) be the semilinear parabolic operator appearing in the statement of the problem. Clearly it is uniformly parabolic. Suppose that \(u,v \in C^2(D) \cap C^1(\overline D)\) are two solutions. Since they agree on the parabolic boundary \(\Sigma\) and satisfy \(\mathcal S(u) = \mathcal S(v) = 0\text{,}\) two applications of Theorem 6.7 yield \(u \le v\) and \(v \ge u\) in \(D\text{,}\) or in other words \(u \equiv v\) in \(D\text{.}\)
2. (PS10) Bounding a solution to a semilinear problem.
Suppose that \(\Omega\) is bounded and that \(u \in
C^2(D) \cap C^1(\overline D)\) satisfies
\begin{align*}
\left\{
\begin{alignedat}{2}
-\partial_t u + \Delta u + u\partial_1 u + \sin u \amp= 0 \amp\qquad\amp \ina D = \Omega \times (0,T],\\
u \amp= 0 \amp\amp \ona \partial\Omega \times (0,T],\\
u \amp= u_0 \amp\amp \ona \Omega \times \{0\}.
\end{alignedat}
\right.
\end{align*}
If the initial data \(u_0=u_0(x)\) satisfies \(0 \le u_0 \le \pi\text{,}\) show that \(u\) satisfies \(0 \le u \le \pi\) in \(D\text{.}\)
Hint.
The constant functions \(0\) and \(\pi\) can be used as comparison functions.
Solution.
The semilinear operator \(\mathcal S(u) = -\partial_t u + \Delta u + u\partial_1 u +
\sin u \) is clearly uniformly parabolic because of the Laplacian, and \(F(x,t,z,p)=zp_1+\sin z\) is certainly \(C^1\text{.}\) Moreover, a quick calculation shows that the constant functions \(0\) and \(\pi\) satisfy
\begin{gather*}
0 \le u \le \pi \ona \Sigma,
\qquad \mathcal S(0) = \mathcal S(\pi) = \mathcal S(u) = 0.
\end{gather*}
