The authors formulate a theory for scalar conservation laws with regularized second and fourth order diffusions. This type of problems is motivated physically by thin liquid films. A new non-classical Riemann solver for concave-convex flux-function is introduced. This solver is based on the kinetic relation, which detremines the speed of under-compressive nonclassical shocks, as well as by a nucleation criterion, which makes a choice between classical and nonclassical Riemann solution. The authors prove the existence of nonclassical entropy solutions of the Cauchy problem and discuss the wave interaction possibilities.