MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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!

share|cite|improve this question
up vote 4 down vote accepted

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.

share|cite|improve this answer
Thank you. Could you please point me to some good references about this topic? – Chong Luo Feb 22 '12 at 0:28
Is there a counter-example that the conclusion is not true if we don't assume this condition? Thanks! – Chong Luo Feb 22 '12 at 0:30
@Chong. See my Edit. – Denis Serre Feb 22 '12 at 13:00
Thanks! That's very helpful. – Chong Luo Feb 22 '12 at 13:07
It seems that this should be for some $\epsilon >0$, not every. – JCM Jun 4 '14 at 13:21

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.