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The non-autonomous wave equation with general Wentzell boundary conditions. (English) Zbl 1071.35098

Summary: We study the problem of the well-posedness for the abstract Cauchy problem associated to the non-autonomous one-dimensional wave equation \(u_{tt}= A(t)u\) with general Wentzell boundary conditions \[ A(t)u(j,t)+ (-1)^{j+1} \beta_j(t){\partial u\over\partial x} (j,t)+ \gamma_j(t) u(j,t)= 0,\quad\text{for }j= 0,1. \] Here \(A(t)u:= (a(x, t)u_x)_x\), \(a(x, t)\geq\varepsilon> 0\) in \([0, 1[\times [0, +\infty)\) and \(\beta_j(t)> 0\), \(\gamma_j(t)\geq 0\), \((\gamma_0(t),\gamma_1(t))\neq(0, 0)\). Under suitable regularity conditions on \(a\), \(\beta_j\), \(\gamma_j\) we prove the well-posedness in a suitable (energy) Hilbert space.

MSC:

35L90 Abstract hyperbolic equations
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