I am currently working on a diffusion problem involving partial differential equations and am a bit confused about how the Laplacian (Laplace?) operator works on different independent variables such as time and space. Here is a picture of the question: PDE Problem involving space and time.
Now what I initially tried to do was expand out the laplace term into a second derivative in x and a second derivative in time. However, in the solutions which are provided in the following picture, they do not expand out a second derivative in time and I am confused as to why they would only do it in x. Solution to PDE. Does the laplace operator only work on spacial variables then? Thank you for your help.
Note the method being used to solve the problem is a separation of variables.
Usually when people write a Laplacian when there is a clear "time" variable and "space" variable(s), the Laplacian only hits the space variables. Sometimes people will explicitly write $\Delta_x$ or $\nabla^2_x$ to denote this, but usually it's clear from context. Here for instance they are talking about diffusion which definitely does not have a second time derivative involved.