Local times for systems of non-linear stochastic heat equations

We consider $$u(t,x)=(u_1(t,x),\cdots ,u_d(t,x))$$ the solution to a system of non-linear stochastic heat equations in spatial dimension one driven by d-dimensional space-time white noise. prove that, when $$d\le 3$$ , local time $$L(\xi ,t)$$ $$\{u(t,x)\,,\;t\in [0,T]\}$$ exists and $$L(\cdot ,t) $$ belongs a.s. Sobolev space H^{\alpha }({\mathbb {R}}^d)$$ for $$\alpha <\frac{4-d}{2}$$ $$d\ge 4$$ does not exist. also show joint continuity establish Hölder conditions . These results are then used investigate irregularity coordinate functions Comparing similar obtained linear equation (i.e., is Gaussian), we believe that our sharp. Finally, get sharp estimate partial derivatives density $$(u(t_1,x)-u(t_0,x),\cdots ,u(t_n,x)-u(t_{n-1},x))$$ which new result independent interest.