Lagrangian for nonlinear perturbed heat and wave equations

The perturbed heat and wave equations [A.H. Bokhari, A.G. Johnpillai, F.M. Mahomed, F.D. Zaman, Approximate conservation laws of nonlinear perturbed heat and wave equations, Nonlinear Analysis. Real World Applications 13 (2012) 2823–2829] are studied, which were considered to admit no standard Lagrangian. By the semi-inverse method, however, an exact Lagrangian is obtained and its proof is given.