Global solution of a nonlinear string equation
The author studies the nonlinear equation \[ w_{tt} (x, t)- \Biggl(\lambda+ {2\over \pi} \int^\pi_0 w^2_x(x, t)dx\Biggr) w_{xx}(x, t)= P(x)\tag{1} \] on the interval \(0\leq x\leq \pi\), with \(\lambda< 0\), given the boundary conditions \(w(0, t)= w(\pi, t)= 0\) and the initial conditions \(w(x, 0)= f(x)\), \(w_t(x, 0)= g(x)\). A solution in the form \(w(x, t)= \sum^\infty_{j= 1} T_j(t)\sin jx\) is searched, and the global existence of a weak solution with \(\lambda< 0\) and \(P\not\equiv 0\) is shown, provided the initial data are sufficiently close to the tensile equilibrium solution of (1). Finally, the author shows that only one such solution exists.
scientific article; zbMATH DE number 811157