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 $t_0$ $$ u(x, t_0) \equiv C, \quad x \in (0, 1). $$

I have a feeling that there should be some contradiction to analyticity argument, but cannot complete a proof.

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

Consider parabolic equation $$ u_t = u_{xx} + f(x, t), $$ where $x \in (0, 1)$, $t \in R^+$, $f \in L^{\infty}$ and Neumann boundary condition. Assume that initial data is smooth and not constant, and $f(x, t)$ is the indictor function of certain measurable set $A \subset (0, 1) \times R^+$.

Is in possible that in finite time $t_0$ there exists an interval $(a, b) \subset (0, 1)$ such that $$ u(x, t_0) \equiv C, \quad x \in (a, b). $$

I have a feeling that there should be some contradiction to analyticity argument, but cannot prove this due to low regularity of $f$.

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 $t_0$ $$ u(x, t_0) \equiv C, \quad x \in (0, 1). $$

I have a feeling that there should be some contradiction to analyticity argument, but cannot complete a proof.

Consider parabolic equation $$ u_t = u_{xx} + f(x, t), $$ where $x \in (0, 1)$, $t \in R^+$, $f \in L^{\infty}$ (but not necessarily continuous) and Neumann boundary condition. Assume that initial data is smooth and not constant.

Is in possible that in finite time $t_0$ there exists an interval $(a, b) \subset (0, 1)$ such that $$ u(x, t_0) \equiv C, \quad x \in (a, b). $$

I have a feeling that there should be some contradiction to analyticity argument, but cannot prove this due to low regularity of $f$.

Consider parabolic equation $$ u_t = u_{xx} + f(x, t), $$ where $x \in (0, 1)$, $t \in R^+$, $f \in L^{\infty}$ and Neumann boundary condition. Assume that initial data is smooth and not constant, and $f(x, t)$ is the indictor function of certain measurable set $A \subset (0, 1) \times R^+$.

Is in possible that in finite time $t_0$ there exists an interval $(a, b) \subset (0, 1)$ such that $$ u(x, t_0) \equiv C, \quad x \in (a, b). $$

I have a feeling that there should be some contradiction to analyticity argument, but cannot prove this due to low regularity of $f$.