The equation is \begin{equation} \left\{\begin{aligned} &\frac{\partial N(t,x)}{\partial t}=\frac{\partial ^2 N(t,x)}{\partial x^2}&x\in \Bbb R\\ &N(0,x)=N_0(x) &x\in \Bbb R\\ &\lim_{x\to +\infty}\frac{\partial N}{\partial x}=\lim_{x\to -\infty}\frac{\partial N}{\partial x}=0. \end{aligned}\right. \end{equation} Is the solution of above equation only a constant? If so, how to prove it?
Could someone help me see the next step? I'm stuck.