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!
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You need essentially the same condition as in the case of the domain $x\in\mathbb R$. That is, $$u(x,t)=o(e^{\epsilonx^2})$$ for every $\epsilon>0$. Edit. Tikhonov provided an example of a nontrivial 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 nontrivial 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. 

