Structure of sign changes under the heat flow

Yes, this is possible (at least if you require only that $$u$$ be defined for $$t>0$$, which is the usual context for the heat equation, not for all $$t \in {\bf R}$$ as implied by "$$u(t,x) \in C^\infty({\bf R} \times {\bf R}^2)$$"):

(source: harvard.edu)

Here $$u(t,x)$$ is antisymmetric about the vertical axis $$x_2=0$$; thus the zero-set $$V_{u(t,\cdot)} = \{x: u(t,x) = 0\}$$ is always symmetric about that axis and contains it. The curved contours of that Sage plot, in black, red, orange, green, blue, purple, and gray, show the other component(s) of $$V_{u,(t,\cdot)}$$ for $$t = t_1/8$$, $$t_1/4$$, $$t_1/2$$, $$t_1$$, $$2t_1$$, $$4t_1$$, $$8t_1$$. The zero set $$V_{u,(t,\cdot)}$$ contains a $$\ast$$-shaped triple point for $$t=t_1$$, and two $$+\,$$-shaped double points for all $$t > t_1$$, at height proportional to $$\pm \sqrt{t-t_1}$$.

To obtain this function, start from the usual heat kernel $$g(t,x) = (4\pi t)^{-1} \exp (-|x|^2/4t)$$, and set $$u(t,x) = \Delta_x(g(t,x-e_1)-g(t,x+e_1))$$ where $$e_1$$ is the unit vector $$(1,0)$$. (To check that $$u$$ is a solution of the heat equation $$u_t = \Delta_x u$$, note that $$g$$ satisfies the heat equation and that the differential operators $$\partial / \partial t$$ and $$\Delta_x$$ commute with $$\Delta_x$$ and with translations by $$\pm e_1$$.) The nodal set of each term $$\Delta_x(g(t,x\mp e_1))$$ of $$u$$ is the circle of radius $$2t^{1/2}$$ about $$\pm e_1$$. For small $$t$$, the nodal set $$V_{u(t,\cdot)}$$ of the difference consists of the vertical axis and very close approximations to those circles. As $$t$$ increases, the circles grow and distort, eventually meeting to form a figure-eight at $$t=t_1$$ and a single closed curve for all $$t > t_1$$. The two double points for $$t>t_1$$ can be located as the zeros on the vertical axis $$x_2=0$$ of the partial derivative of $$u(t,x)$$ with respect to $$x_1$$.