ENERGY DECAY FOR THE DAMPED WAVE EQUATION ON AN UNBOUNDED NETWORK

We study the wave equation on an unbounded network of N, N is an element of N*, finite strings and a semi-infinite one with a single vertex identified to 0. We consider continuity and dissipation conditions at the vertex and Dirichlet conditions at the extremities of the finite edges . The dissipation is given by a damping constant alpha > 0 via the condition Sigma(N)(j=0)partial derivative(x)u(j)(0, t) = alpha partial derivative(t)u(0)(0, t). We give a complete spectral description and we use it to study the energy decay of the solution. We prove that for alpha not equal N +1 we have an exponential decay of the energy and we give an explicit formula for the decay rate when the finite edges have the same length.