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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.
##### MSC:
 35L70 Second-order nonlinear hyperbolic equations 35C20 Asymptotic expansions of solutions to PDEs 35L20 Initial-boundary value problems for second-order hyperbolic equations
##### Keywords:
multiple scale; perturbation
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