On boundary value problem for a class of retarded nonlinear partial differential equations (Q5930139)

The author investigates a problem of nonlinear oscillations of an elastic plate in a potential supersonic gas flow. A quasistatic case is considered, i.e., when the inertial forces are essentially weaker than the resonance forces. This problem is described by a class of retarded quasilinear partial differential equations NEWLINE\[NEWLINE \gamma \dot u+\triangle^2u-f\Biggl(\int\limits_{\Omega } \nabla u(x,t) ^2 dx\Biggr)\triangle u+\rho \partial u/\partial x_{1}-q(u_t)=d_0(x), \;x\in\Omega, \;t>0 NEWLINE\]NEWLINE with the boundary condition \(u _{\partial \Omega }=\triangle u _{\partial \Omega }\). Here \(\Omega \) is a bounded domain in \(\mathbb{R}^2\), \(x=(x_1,x_2)\), \(\gamma \), \(\rho \) are positive parameters of the system. The retarded term \(q(u_t)\) has a special form. The main result is that there exists a strong solution of the considered problem in any interval \([0,T]\) under some assumptions, for instance, \(f\) is local Lipschitz and satisfies the inequality \(\varliminf_{s\to\infty }f(s)\geq -C_f\) (\(C_f\) is a constant). Moreover, the solution is unique and in the class \(L^2\).