This paper is devoted to closed-form solutions of the partial differential equation: θ x x + θ y y + δ exp ( θ ) = 0 , which arises in the steady state thermal explosion theory. We find simple exact solutions of the form θ ( x , y ) = Φ ( F ( x ) + G ( y ) ) , and θ ( x , y ) = Φ ( f ( x + y ) + g ( x - y ) ) . Also, we study the corresponding nonlinear wave equation.