Stabilization and asymptotic behavior of a generalized telegraph equation
The author studies the following generalization of the telegraph equation: \[ \begin{cases} V_t + gV + aI_x + kW = 0 & \text{in } (0,L) \times (0, \infty), \\ I_t + rI + b V_x = 0 & \text{in } (0,L) \times (0,\infty), \\ W_t + cW = V & \text{in } \Omega \times (0, \infty), \\ V(0,\cdot) = V(L,\cdot) = 0 & \text{on } (0, \infty), \\ V(\cdot,0) = V_0, I(\cdot,0) = I_0, W(\cdot,0) = W_0, & \text{in } (0,L) \end{cases} \] for some \(L^\infty\) coefficients which are additionally assumed to satisfy \(a \geq \delta\), \(b \geq \delta\), \(c \geq \delta\) and \(k+g \geq \delta\) for some \(\delta > 0\). It is shown that for a suitable energy norm the realization of the above problem generates a \(C_0\)-semigroup. Further, the strong stability of the semigroup is obtained with the help of the ABVL theorem. Afterwards, the decay rates of the solutions are investigated in dependence of the coefficients. The author establishes polynomial decay and under some additional assumptions even exponential decay of the solutions. In the special case of constant coefficients the optimality of the decay rates is shown.
scientific article; zbMATH DE number 6526491
scientific article