Systems of nonlinear wave equations with nonlinear viscosity

Avner Friedman and Jindrich Necas

Vol. 135 (1988), No. 1, 29–55

DOI: 10.2140/pjm.1988.135.29

Abstract

An equation of the form

where p = ∇u, q = ∇u, u = ∂u∕∂t, ü = ∂²u∕∂t² represents, for suitable functions W(p), V (q), a nonlinear hyperbolic equation with nonlinear viscosity and it appears in models of nonlinear elasticity. In this paper existence and regularity of solutions for the Cauchy problem will be established. In particular, if n = 2, or if n ≥ 3 and the eigenvalues of (∂²V∕∂q_j∂q_j) belong to a “small” interval, then the solution is classical. These results will actually be established for a system of equations of the above type.

Mathematical Subject Classification 2000

Primary: 35L70

Secondary: 35K55, 73F99

Milestones

Received: 24 April 1987

Published: 1 November 1988