# Strong maximum principle for heat equation; If $u(x,t)$ is constant for times $t\leq t_0$, is $u$ also constant at times $t\geq t_0$?

I am learning the strong maximum principle by following Evan's Partial differential equations. It is stated as follows: Let $$U$$ be an open subset of $$\mathbb{R}^n$$ and $$T>0$$. Let $$u=u(x,t)$$, where $$x=(x_1,\ldots,x_n)\in\Omega$$ and $$0\leq t\leq T$$. Denote $$\partial_t$$ the partial derivative wrt $$t$$ and $$\Delta_x=\sum_{i=1}^n\partial_{x_i}^2$$ the Laplacian on the variables $$x_i$$.

Theorem 4(b) (page 55) of Evan's book states, in equivalent terms

Theorem (strong maximum principle for heat equation): Suppose that

• $$U$$ is connected;
• $$u$$ is $$C^2$$ on the variables $$x_i$$, and $$C^1$$ on the variable $$t$$, on $$U\times(0,T]$$, and continuous on all of $$\overline{U}\times [0,T]$$;
• (the heat equation) $$\partial_t u=\Delta_x u\quad\text{on}\quad U\times (0,T)$$
• There exists $$x_0\in U$$ such that $$u(x_0,T)=\sup u(U\times([0,T]))$$

Then $$u$$ is contant of $$\overline{U}\times[0,T]$$.

My question is: Does the analogous result hold if we assume instead that $$u$$ attains its maximum at a point $$(x_0,t_0)$$, where $$0?

My guess is that this is not the case: If there is a nonzero solution $$v$$ of the heat equation on some square $$U\times[0,T]$$ which satisfies $$v=0$$ when $$t=0$$, then we can glue the constant function $$0$$ on $$U\times[0,t_0]$$ with $$w(x,t)=v(x,t-t_0)$$ when $$t\geq t_0$$, and hope that the resulting function is sufficiently regular. However we should not have any good boundedness assumptions on $$v$$ to avoid all those uniqueness theorems for solutions of the heat equation..

You are right. If $$u$$ attains its maximum at $$(x_0,t_0)$$, then you can only conclude that $$u$$ must be constant up to that time-point, i.e. in $$\bar{U}_{t_0}$$. Beyond $$t_0$$, the function may change value before reaching the final time $$T$$.