Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem Page: 2
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Unsurangsie, Sumalee, Existence of a Solution for a Wave
Equation and an Elliptic Dlrichlet Problem. Doctor of
Philosophy (Mathematics), May, 1988, 59 pp., bibliography, 19
titles.
In this paper we consider an existence of a solution for
a nonlinear nonmonotone wave equation in [0,t]xR and an
existence of a positive solution for a non-positone Dirichlet
problem in a bounded subset of Rn.
For a wave equation
- u + /lu = f(x,t) + g(u(x, t)) ,
L T. <(Uv
u(0, t) = U(5T,t) = 0, u(X,t) = U(X,t+2X ) ,
2
we
show that if f 38 cq + r, where q and r € L and
/I € R - {j2 - k2:k = 1,2 j = 0,1,...}, then there exists
cQ such that if |c| > cQ the wave equation has a weak
solution. The solution is obtained using an iteration
argument.
For the elliptic Dirichlet problem
-JAu = f(u) in ft,
u = 0 on <?ft,
where A is the Laplace operator, and ft is a smooth bounded
domain in Rn, we show that if X >0, f is a nondecreasing
function such that f(x) = f(0) < 0 for x < 0, and f satisfies
some additional conditions, then there exists /lQ > 0 such
that if 0 < \ < Jq the Dirichlet problem has a positive
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Sumalee Unsurangsie. Existence of a Solution for a Wave Equation and an Elliptic Dirichlet Problem, dissertation, May 1988; Denton, Texas. (https://digital.library.unt.edu/ark:/67531/metadc331780/m1/2/: accessed March 29, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; .