Gronwall's Inequality for Systems of Partial Differential Equations in Two Independent Variables

This paper presents a generalization for systems of partial differential equations of Gronwall's classical integral inequality for ordinary differential equations. The proof is by reducing the vector integral inequality to a vector partial differential inequality and then using a vector generalization of Riemann's method to obtain the final inequality. The final inequality involves a matrix function in the integrand which is a generalization of the scalar Riemann function. The proof includes a successive approximations argument to guarantee the existence and positivity property of this matrix function. The inequality is applied to prove a uniqueness theorem for a nonlinear vector hyperbolic partial differential equation, a comparison theorem for a linear hyperbolic vector partial differential equation, and a continuous dependence theorem for a nonlinear vector boundary value problem. The inequality also appears to have many applications instability problems and in numerical solutions of partial differential equations. All of these results hold for the corresponding Volterra integral equations and the method of proof of the main result shows that the function on the right-hand side of the final inequality is the solution of the integral equation and hence is the maximal solution of the original inequality.