Let $u(x, t)$ be a solution of $u\_t=u\_{xx}$ in the domain $x>0, t>0$. We also have the initial condition $u(x,0)=g(x)$ and the boundary condition $u(0,t)=h(t)$. Do we have maximum principle in this case? Can we conclude that $u(x,t)$ is bounded if we assume both $g$ and $h$ are bounded? If not, what additional condition shall we impose? Thanks!



Maximum principle for heat equation on infinite domain
You need essentially the same condition as in the case of the domain $x\in\mathbb R$. That is, $$u(x,t)=o(e^{\epsilon|x|^2})$$
for every $\epsilon>0$.


**Edit**. Tikhonov provided an example of a non-trivial solution of the heat equation on the domain $\mathbb R$, with zero data. Take either its odd part, or the derivative of its even part with respect to $x$. It is a non-trivial solution of the heat equation in the domain $(0,+\infty)$ with zero Dirichlet boundary condition and zero initial data. If such a principle as the one considered by the MO author existed, this solution would be trivial.


