Problem: Consider the PDE $\u_t + \frac{2x}{t} u_x = u_{xx} + (1-u^2) \sinh(x), \quad u(x,1) = \frac{1}{4}e^{-x^2} -1, \lim_{|x| \to \infty} u(x, t) = -1$. Find the solution in the limit $t \to \infty$. Use only the variables and constants given in the problem; do not define additional constants. Place your final solution in a $\boxed{}$ LaTeX environment. Do not define additional parameters or constants.
Solution: $\boxed{ u(x,t) = \tanh(-\cosh(x) + x^2/t)}$