Cauchy partial differential equation

prove d/dt(u(x(t),t))+tanh(x(t))(d/dx(u(x(t),t)))=0
           u(x(t),0)=a(x(t))
           limit as x tends to infinity of u(x,t)=0
  
has at most one solution.Explain why there is no boundary condition at x=0
and find  a solution for the special case a(x(t))=sinh(x(t))