Vol. 2, No. 2, 2009

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The high exponent limit $p \to \infty$ for the one-dimensional nonlinear wave equation

Terence Tao

Vol. 2 (2009), No. 2, 235–259

We investigate the behaviour of solutions ϕ = ϕ(p) to the one-dimensional nonlinear wave equation ϕtt + ϕxx = |ϕ|p1ϕ with initial data ϕ(0,x) = ϕ0(x), ϕt(0,x) = ϕ1(x), in the high exponent limit p (holding ϕ0,ϕ1 fixed). We show that if the initial data ϕ0,ϕ1 are smooth with ϕ0 taking values in (1,1) and obey a mild nondegeneracy condition, then ϕ converges locally uniformly to a piecewise limit ϕ() taking values in the interval [1,1], which can in principle be computed explicitly.

nonlinear wave equation
Mathematical Subject Classification 2000
Primary: 35L15
Received: 22 January 2009
Revised: 13 February 2009
Accepted: 5 April 2009
Published: 1 May 2009
Terence Tao
Department of Mathematics
University of California, Los Angeles
Mathematics Department
Los Angeles, CA 90095-1555
United States