Laplace transform to bio heat equation This is the bio heat equation  and I have several questions when trying to work with it.
$$
 \rho c \frac{\partial u(x,t)}{\partial t}
= \nabla[k \nabla u(x,t)]
  + \omega_b \rho_b c_b [u_a - u(x,t)]
  + Q_m + Q_r(x,t)
$$
where the first expression on the right hand side describes the conduction of heat induced by temperature gradient.
Then by letting $k$ to be a constant through out the process this equation has been written as  
$$
 \rho c \frac{\partial u(\mathbf{x},t)}{\partial t}
 = k \nabla^2 u(\mathbf{x},t)
   +\omega_b \rho_b c_b[ u_a - u(\mathbf{x},t) ]
   + Q_m + Q_r(\mathbf{x},t).
$$ 


*

*Here does $\nabla[k \nabla u(x,t)]$ become $k \nabla^2 u(\mathbf{x},t)$ due to this constant $k$?     

*When taking the Lapalace transform what is the Laplace transform of the term $k \nabla^2 u(\mathbf{x},t)$?

*The bio heat equation at the top of the post is for a 2-D case. That is $\mathbf{x} = (x_1,x_2)$. But if this equation is written for a 1-D case at steady state temperature does the bio heat equation become
$$
 pc \frac{\partial u(x)}{\partial t}
 = \frac{\partial^2 u(x)}{\partial^2 x}
   + \omega_b p_b c_b[ u_a - u(x) ]
   +Q(m) + Q(x)?
$$

*

*I think yes if $$\nabla[k \nabla u(\underline x,t)] = \nabla \cdot [k \nabla u(\underline x,t)]:$$


By one of the product rules (),
$$\nabla[k \nabla u(\underline x,t)]$$
$$= \nabla k \nabla u(\underline x,t) + k \nabla \nabla u(\underline x,t)]$$
$$= (0) \nabla u(\underline x,t) + k \nabla \nabla u(\underline x,t)]$$
$$= (0) \nabla u(\underline x,t) + k \nabla \nabla u(\underline x,t)]$$
$$= k \nabla \nabla u(\underline x,t)]$$
But what does $$\nabla \nabla u(\underline x,t)$$
even mean?
By definition, $\nabla^2 u(\underline x,t) = \nabla \cdot \nabla u(\underline x,t)$.
Also for 'well-behaved' u(\underline x,t), we have:
$$\nabla \times \nabla u(\underline x,t) = 0$$



*Try (10) and (11) here.







*Steady state temperature means $t = 0$?


If so, I think so except I think the last terms should be: $Q_m + Q_r(x)$ based on (1) and (4) here. In the link, it seems that $Q_r(x) = 0$




