Gronwall's inequality for systems of partial differential equations in two independent variables

Author:
Donald R. Snow

Journal:
Proc. Amer. Math. Soc. **33** (1972), 46-54

MSC:
Primary 35B45

DOI:
https://doi.org/10.1090/S0002-9939-1972-0298188-1

MathSciNet review:
0298188

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Abstract: 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 in stability 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.

**[1]**D. R. Snow,*A two-independent variable Gronwall-type inequality*, Third Sympos. on Inequalities (Los Angeles, September 1969), Proc. Sympos. on Inequalities, Academic Press, New York, 1971, pp. 330-340. MR**0338537 (49:3301)****[2]**T. H. Gronwall,*Note on the derivatives with respect to a parameter of the solutions of a system of differential equations*, Ann. of Math. (2)**20**(1919), 292-296. MR**1502565****[3]**R. Bellman,*The stability of solutions of linear differential equations*, Duke Math. J.**10**(1943), 643-647. MR**5**, 145. MR**0009408 (5:145a)****[4]**W. Walter,*Differential- und Integral-Ungleichungen und ihre Anwendung bei Abschätzungs- und Eindeutigkeitsproblemen*, Springer Tracts in Natural Philosophy, vol. 2, Springer-Verlag, Berlin, 1964, pp. 125-126; English transl., Springer-Verlag, Berlin, 1970, pp. 143-144. MR**30**#2302. MR**0172076 (30:2302)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1972-0298188-1

Keywords:
Gronwall's inequality,
differential inequality,
integral inequality,
hyperbolic systems,
system of Volterra integral equations,
uniqueness theorems,
comparison theorems,
continuous dependence,
stability,
numerical solution,
vector characteristic initial value problems,
Riemann's method,
Riemann functions,
maximal solutions,
matrix Lipschitz condition

Article copyright:
© Copyright 1972
American Mathematical Society