Vol. 32, No. 2, 1970

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A uniqueness theorem for second order quasilinear hyperbolic equations

Albert Emerson Hurd

Vol. 32 (1970), No. 2, 415–427
Abstract

A uniqueness theorem is proved for weak solutions of quasilinear second-order hyperbolic equations of the form

     ∑n -∂- ˙t
utt −   ∂xia (x,t,u,u1⋅⋅⋅ ,un) = b(x,t,u)
i=1

in many space variables. The weak solutions are assumed to satisfy a time-wise upper Lipschitz bound

uk(x,t1)−-uk(x,t2)
t1 − t2      ≦ K (t)

for all 0 < t t1,t2 where K(t) is an L1-function. Together with the obvious assumptions, the equation is supposed to satisfy a symmetry condition

  i     j
∂a--= ∂a--
∂uj   Θui

along with convexity of the ai in u and uk. As a corollary, a uniqueness theorem for systems proved by Oleinik is generalized.

Mathematical Subject Classification
Primary: 35.57
Milestones
Received: 13 September 1968
Revised: 30 June 1969
Published: 1 February 1970
Authors
Albert Emerson Hurd