2
$\begingroup$

Consider parabolic equation $$ u_t = u_{xx}, $$ where $x \in (0, 1)$, $t \in (0, T)$ with nonhomogenious Dirichlet boundary conditions. $$ u(0, t) = \psi_0(t) \in C^0, $$ $$ u(1, t) = \psi_1(t) \in C^0. $$

Assume that initial data $$ u(x, 0) = \varphi(x) $$ is smooth and not constant,

Is it possible that in finite time $t_0$ $$ u(x, t_0) \equiv C, \quad x \in (0, 1). $$

ANSWER: Yes, it can

Improve this question
$\endgroup$
4
  • 1
    $\begingroup$ As it stands, the answer is obviously yes. Just pick u to be whatever you want it to be and then choose f accordingly. But this is probably not what you intended. $\endgroup$
    – Michael Renardy
    Commented Sep 26 at 1:00
  • 2
    $\begingroup$ Yes, indeed. I have some restrictions on $f$. It is an indicator of a certain set in $(x, t)$. I am editing question. $\endgroup$
    – Sergey Tikhomirov
    Commented Sep 26 at 1:10
  • $\begingroup$ It seems that Feymann-Kac formula helps with backward solution of $u_t = -u_{xx}$, which is a lot different. $\endgroup$
    – Sergey Tikhomirov
    Commented Sep 27 at 19:49
  • $\begingroup$ @SergeyTikhomirov it works for both, see the answer in the above link. $\endgroup$
    – Thomas Kojar
    Commented Sep 27 at 19:58

1 Answer 1

Reset to default
5
$\begingroup$

Not only is this possible, but more is true: For any initial datum, there exist boundary data which make it so. Google "boundary null controllability" for the heat equation.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Thank you. Yes it works. $\endgroup$
    – Sergey Tikhomirov
    Commented Sep 30 at 0:16

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .