Springer Nature is making Coronavirus research free. View research | View latest news | Sign up for updates

On a doubly damped quasilinear wave equation

  • 30 Accesses


We prove global existence, uniqueness, and asymptotic decay of solutions for an initial and boundary value problem for the equation

$$u_{tt} - u_{xx} - \frac{\partial }{{\partial x}}\beta (u_{xt} ) + f(u_t ) = g(x,t),(x,t) \in (0,1) \times R^ + ,$$

with Β′ possibly degenerate andf not necessary monotone or Lipschitzian, assuming a smallness condition on the initial data and on the forcing term g.


  1. [1]

    G. Andrews,On the existence of solutions to the equation u tt =u xxt +σ(u x ) x , J. Diff. Eqns.,35 (1980), pp. 200–231.

  2. [2]

    G. Andrews -J. Ball,Asymptotic behavior and changes of phase in one-dimensional non-linear viscoelasticity, J. Diff. Eqns.,44 (1982), pp. 306–341.

  3. [3]

    J. C. Clements,On the existence and uniqueness of solutions of the equation u tt − −(∂/∂x i )σ i x i )−δ N u t =f, Canad. Math. Bull.,18 (1975), pp. 181–187.

  4. [4]

    C. M. Dafermos,The mixed initial boundary value problem for the equation of nonlinear one-dimensional viscoelasticity, J. Diff. Eqns.,6 (1959), pp. 71–86.

  5. [5]

    Dang Dinh Hai,On a strongly damped quasilinear wave equation, Demonstratio Math.,19 (1986), pp. 327–340.

  6. [6]

    Dang Dinh Hai,On a quasilinear wave equation with nonlinear damping, Proc. Roy. Soc. Edinburgh,110 A, (1988), pp. 227–239.

  7. [7]

    J. M. Greenberg,On the existence, uniqueness and stability of solutions of the equation ρ0 μ tt =E(X x )X xx X xxt , J. Math. Anal. Appl.,25 (1969), pp. 575–591.

  8. [8]

    J. M. Greeberg -R. C. MacCamy -V. J. Mizel,On the existence uniqueness and stability of solutions of the equation σ′ x xx +λμ xtx 0μ tt , J. Math. Mech.,17 (1968), pp. 707–728.

  9. [9]

    J. L. Lions,Quelques méthode de résolution de problème aux limites nonlinéaires, Dunod, Gauthier-Villars, Paris (1969).

  10. [10]

    M. Prestel,Forced oscillations for the solutions of nonlinear hyperbolic equation, Nonlinear Analysis,6 (1982), pp. 209–216.

  11. [11]

    Y. Yamada,Quasilinear wave equations and related nonlinear evolution equation, Nagoya Math. J.,84 (1981), pp. 31–83.

Download references

Author information

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Hai, D.D. On a doubly damped quasilinear wave equation. Annali di Matematica pura ed applicata 160, 77–87 (1991).

Download citation


  • Initial Data
  • Wave Equation
  • Global Existence
  • Force Term
  • Smallness Condition