A partial differential equation with boundary and initial conditions

Question

Suppose that we have a partial differential equation with boundary and initial conditions.

1st Question

If one of the boundary conditions of PDE is $u(0,t)=0$, can we say $u_t(0,t)$ is always equal to zero?

2nd Question

If one of the initial conditions of PDE is $u(x,0)=\text{sech}^2(10x)$, can we guess about $u_x(x,0)$? For example, can we say that $u_x(x,0)=0$ or $u_x(x,0)\neq0$ ?

Thank you

Answer 1

The answer of the first question is yes. Differentiating $u(0,t)=0$ with respect to $t$ gives $\partial_t u(0,t) = 0$.

For the second question, if one has $u(x,0) = \mathrm{sech}^2(10x)$, then differentiating with respect to $x$ gives $$\partial_x u(x,0) = \frac{d}{dx} [\mathrm{sech}^2(10x)]=-20 \,\mathrm{sech}^2(10x)\, \mathrm{tanh}(10x)$$ In particular, $\partial_x u(x,0) = 0$ only for $x = 0$.