Let $v\in L^p(\Bbb R^d)$, $1\leq p<\infty$ be nonzero function, i.e., $v\not\equiv 0$. Define $$u(x)= |v|*\phi(x)= \int_{\Bbb R^d} |v(y)|\phi(x-y)d y$$ with $\phi(x)= ce^{-|x|^2}$ and $c>0$ so that $\|\phi\|_{L^1(\Bbb R^d)}=1.$
Clearly we have $u\in L^p(\Bbb R^d)\cap L^\infty(\Bbb R^d)\cap C^\infty(\Bbb R^d)$, $u(x)>0$ for all $x\in \Bbb R^d$ and $u(x)\to 0$ as $|x|\to\infty$.
Question:
1- Is it possible that $u$ is nowhere locally constant, that is, the following does not hold:
$$\exists a\in \Bbb R^d, r>0 \quad \text{such that $u(x)= u(a)$ for all $x\in B(a,r)$?}$$
2- If not, can we find $n\geq1$ such that $u_n$ fullfils the first question? Here we put $u_1= u$ and $u_n=u_{n-1}*\phi$.