Translation of weak solution again a weak solution?

Question

I am currently struggling with the question the title says. To give an example: let us consider the weak formulation of the Heat equation $$\displaystyle\iint_{\Omega_T}-u\partial_t\phi + \nabla u \cdot \nabla \phi dx dt =0$$ for any smooth test function with compact support in $\Omega_T$, where $\Omega_T$ is a usual space-time cylinder. How can I translate the weak solution $u$ so that it still is a weak solution? I would like some to have that $v(x,t) = \frac{u(\hat{x}+ax,\hat{t}+b^\alpha a^\beta t)}{b}$ is again a weak solution (at least locally, so in some compactly contained subset of $\Omega_T$) where the parameters $\alpha,\beta$ are to be chosen correctly ($\hat{x},\hat{t}$ are some fixed points in $\Omega_T$). I have recently seen this idea for more general doubly nonlinear pde $\partial_t u^r - \Delta_p(u)=0$ where $r>0,p>1$ but I just can't figure out how this scaling works and why their scaled function (with the exponents $\alpha = r+1-p, \beta = p$) is again a weak solution (locally). Thanks for any help!