Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions
- 426 Downloads
The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time.
KeywordsWave equation Dynamic boundary condition Source Damping Blow up Finite time Exponential growth
The first author wishes to thank the Virginia Space Grant Consortium and the Jefferson Scholars Foundation for their support. The second author wants to thank KAUST for its support. Both authors are very grateful to Prof. Irena Lasiecka for many fruitful discussions.
- 9.Budak, B.M., Samarskii, A.A., Tikhonov, A.N.: A Collection of Problems on Mathematical Physics. Macmillan Co., New York (1964). Translated by A.R.M. Robson Google Scholar
- 10.Caroll, R.W., Showalter, R.E.: Singular and Degenerate Cauchy Problems. Academic Press, New York (1976) Google Scholar
- 27.Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Springer, New York (1972) Google Scholar
- 42.Yu, S.: On the strongly damped wave equation with nonlinear damping and source terms. Electron. J. Qual. Theory Differ. Equ. 39, 1–18 (2009) Google Scholar