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Asymptotic theory for weakly nonlinear wave equations in semi-infinite domains. (English) Zbl 1036.35140
The author studies the existence and uniqueness of solutions of a class of weakly nonlinear wave equations of form \[ u_{tt}-u_{xx}+\varepsilon h(u,u_t,u_x)=0 \quad (t,x>0, \;0<\varepsilon \ll 1) \] under the initial condition \(u(x,0)=a(x)\), \(u_t(x,0)=b(x)\) and boundary condition \(u(0,t)=\rho (t)\), \(t,x>0\). The solutions are considered in a semi-infinite region \(0\leq x\), \(t<L/\sqrt{| \varepsilon | }\). It is supposed that \(a(x)\), \(\rho (t)\) are twice continuously differentiable for \(x\geq 0\), \(t\geq 0\), \(b(x)\) is continuously differentiable for \(x\geq 0\), \(h\) and its derivatives are analytic and uniformly bounded in its arguments, \(a(0)=\rho (0)\), \(b(0)=\rho^{\prime } (0)\), \(\rho (0)=0\), \(a^{\prime }(0)=0\), \(\rho^{\prime \prime } (0)=0\), \(-a^{\prime \prime }(0)+\varepsilon h(a(0),b(0),a^{\prime }(0))=0\).
Taking into account these conditions the author proves that there exists a unique, twice continuously differentiable solution in a region of the \(x\)-\(t\) plane, \(0\leq x\), \(t\leq L/\sqrt{| \varepsilon | }\), where \(L>0\) is a sufficiently small constant. Moreover, this solution depends continuously on the initial-boundary data. The asymptotic validity of formal perturbation approximations of the solutions is shown in the same region as well.
35L70 Second-order nonlinear hyperbolic equations
35C20 Asymptotic expansions of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
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