Nonlinear boundary value problem for the damped wave equation (Q2713897)

The authors study solvability of the boundary value problem NEWLINE\[NEWLINE Lu=u_{tt} - a(u) u_{xx} +\alpha |u_t|^{\rho}u_t = f(x,t)\;\;((x,t)\in Q=(0,1)\times (0,T)), \tag{1} NEWLINE\]NEWLINE NEWLINE\[NEWLINE (u_x +K(u)u_{tt} +|u_t|^{\rho}u_t)(1,t),\;u(0,t)=0,\;u(x,0)=u_t(x,0)=0\;(x\in (0,1)). \tag{2} NEWLINE\]NEWLINE Here \(\rho, \beta>1\) and \(\alpha>0\) are constants and the functions \(K(u)\) and \(a(u)\) meet the following conditions: NEWLINE\[NEWLINE a(u),K(u)\geq \delta>0,\;K(u)\leq c_0(1+|u|^{\beta}),\;|a'(u)/a(u)|\leq c_1,\;|K'(u)|^{\beta/(\beta-1)}\leq c_2(1+K(u)), NEWLINE\]NEWLINE where \(c_i,\delta\) are positive constants. It is established that, for \(f,f_t\in L_2(Q)\), there exists a unique solution to the problem (1), (2) such that \(u\in L_{\infty}(0,T;W_2^2(0,1))\), \(u_t\in L_{\infty}(0,T;W_2^1(0,1))\) \(u_{tt}\in L_{\infty}(0,T;L_2(0,1))\), and \(u_{tt}(1,t)\in L_\infty(0,T)\). To prove the result, the authors apply the Faedo-Galerkin method.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00039].