Linear Or nonlinear Black Scholes Equation

I have been going through the analytical solutions of black scholes equation which transforms it to a heat equation. $$u_{t}=\frac{1}{2}\sigma^{2}u_{xx}$$ Now if the volatility is constant , then its the linear form. and if the volatility is variable, then its the nonlinear form ? Please give reference too with the answer if possible.

In your example, the dependent variables and its derivative are $$u$$, $$u_t$$ and $$u_{xx}$$. As the equation is linear in all of them, the coefficient don't depend on u or its derivatives, and there are no cross terms (e.g., $$u \, u_x$$), so the equation is linear. The criteria you have listed - constant vs variable volatility - will help differentiate whether the equation is constant-coefficent. On the other hand, if volatility depends on u or its derivatives, then that would make it non-linear.