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On a system of nonlinear wave equations with the Kirchhoff-Carrier and Balakrishnan-Taylor terms. (English) Zbl 1513.35373

The paper deals with the initial-boundary value problem for the system \[u_{tt}-\lambda u_{xxt}-\mu_1(t,\int_0^1u_x(x,t)u_{xt}(x,t)\, dx) u_{xx}=f_1(x,t,u,v,u_x,v_x,u_t,v_t)\] \[v_{tt}-\mu_2(t,\int_0^1v^2(x,t)\, dx,\int_0^1v_x^2(x,t)\, dx)v_{xx}=f_2(x,t,u,v,u_x,v_x,u_t,v_t)\] in \((0,1)\times(0,T)\) with the boundary conditions \(u(0,t)=v_x(0,t)-\zeta v(0,t)=u(1,t)=v(1,t)=0\) and some initial data. In the first part, applying the Faedo-Galerkin method and the weak compact method, the authors prove the local existence and the uniqueness of the weak solution to the problem. In the second part, they construct a suitable Lyapunov functional to obtain the exponential decay of weak solutions.

MSC:

35L57 Initial-boundary value problems for higher-order hyperbolic systems
35Q74 PDEs in connection with mechanics of deformable solids
35D30 Weak solutions to PDEs
35B40 Asymptotic behavior of solutions to PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations
37B25 Stability of topological dynamical systems
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