Solvability of the Cauchy problem for a system of non-linear parabolic equations

Abstract

We study the existence and uniqueness of non-negative smooth bounded solutions to the Cauchy problem for a system of non-linear parabolic equations with monotone and concave non-linearities. Under certain conditions on the non-linear terms and the coefficients of the system, a constructive theorem ensuring the existence of such a solution is established. In addition, an equivalent system of non-linear Hammerstein-Volterra type integral equations is considered. For this system, a special iterative scheme is proposed, which, under an additional condition on the non-linearities, is shown to uniformly converge as geometric progression to a smooth solution of the original Cauchy problem. A solution is shown to be unique in the class of bounded smooth vector functions with non-negative components. Such non-linear parabolic systems are interesting not only in mathematics, but also in biochemistry and chemical kinetics. The paper concludes with applied examples satisfying all the assumptions of our theoretical results.