Two dimensional heat problem with homogeneous Neumann boundary conditions

Question

Let \begin{align*} u_t&=u_{xx}\qquad (x,t)\in [0,\infty) \times (0,\infty)\\ \text{IC}\qquad u(x,0)&=g(x) \\ \text{BC}\qquad u_x(0,t)&=0 \\ \end{align*} I know the solution of this one dimensional heat problem with homogeneous Neumann boundary conditions is given by \begin{equation} u(x,t)=\frac{1}{\sqrt{4\pi t}}\int_{0}^{\infty} \bigg[exp\bigg(-\frac{(x-y)^2}{4t}\bigg)+exp\bigg(-\frac{(x+y)^2}{4t}\bigg)\bigg]\,g(y) \, dy \end{equation} I need help to to have a solution formula to the same problem but in two dimensional. That is, the solution of this problem \begin{align*} u_t&=u_{xx}+u_{yy}\qquad (x,y,t)\in [0,\infty) \times [0,\infty) \times (0,\infty)\\ \text{IC}\qquad u(x,y,0)&=g(x,y) \\ \text{BC}\qquad u_x(0,y,t)&=0, \qquad u_y(x,0,t)=0\\ \end{align*}