Assuming you mean the equation $\kappa \frac{\partial^2u}{\partial x^2}=C_p\frac{\partial u}{\partial t}$, then they are all meaningful. In that case $u(x,t)$ is the temperature at point $x$ and timr $t$.

So $\frac{\partial u }{\partial t}$ represent the variation of temperature at a particular position and time, with respect to time. Thus, it represents a cooling or a heating, depending on its sign.

Assuming you mean the equation $\kappa \frac{\partial^2u}{\partial x^2}=C_p\frac{\partial u}{\partial t}$, then they are all meaningful. In that case $u(x,t)$ is the temperature at point $x$ and time $t$.

So $\frac{\partial u }{\partial t}$ represent the variation of temperature at a particular position and time, with respect to time. Thus, it represents a cooling or a heating, depending on its sign. In response to your comment, yes it can be said that it is the rate of change of temperature with respect to time (and not position!).

Assuming you mean the equation $\kappa \frac{\partial^2u}{\partial x^2}=C_p\frac{\partial u}{\partial t}$, then they are all meaningful. In that case $u(x t)$ is the temperature at point $x$ and timr $t$.

Assuming you mean the equation $\kappa \frac{\partial^2u}{\partial x^2}=C_p\frac{\partial u}{\partial t}$, then they are all meaningful. In that case $u(x,t)$ is the temperature at point $x$ and timr $t$.

So $\frac{\partial u }{\partial t}$ represent the variation of temperature at a particular position and time, with respect to time. Thus, it represents a cooling or a heating, depending on its sign.