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Asymptotics for the modified witham equation
Local and global existence of solutions to a strongly damped wave equation of the $ p $-Laplacian type
Department of Mathematics, University of Nebraska-Lincoln, 203 Avery Hall, Lincoln, NE 68588-0130, USA |
$ p $ |
$u_{tt} - Δ_p u -Δ u_t = 0$ |
$ \Omega \subset \mathbb{R}^3 $ |
$ \Gamma = \partial \Omega $ |
$ Δ_p $ |
$ 2<p<3 $ |
$ p$ |
$ f(u) $ |
$ {W^{1,p}}\left( \Omega \right) $ |
$ L^2(\Gamma) $ |
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
K. Agre and M. A. Rammaha,
Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations, 19 (2006), 1235-1270.
|
[3] |
P. Aviles and J. Sandefur,
Nonlinear second order equations with applications to partial differential equations, J. Differential Equations, 58 (1985), 404-427.
|
[4] |
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010. |
[5] |
A. Benaissa and S. Mokeddem,
Decay estimates for the wave equation of p-Laplacian type with dissipation of m-Laplacian type, Math. Methods Appl. Sci., 30 (2007), 237-247.
|
[6] |
A. C. Biazutti,
On a nonlinear evolution equation and its applications, Nonlinear Anal., 24 (1995), 1221-1234.
|
[7] |
L. Bociu and I. Lasiecka,
Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Appl. Math. (Warsaw), 35 (2008), 281-304.
|
[8] |
L. Bociu and I. Lasiecka,
Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860.
|
[9] |
L. Bociu and I. Lasiecka,
Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.
|
[10] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible NAvierSTokes Equations and Related Models, Springer, 2013. |
[11] |
M. M. Cavalcanti and V. N. Domingos,
Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459.
|
[12] |
F. Chen, B. Guo and P. Wang,
Long time behavior of strongly damped nonlinear wave equations, J. Differential Equations, 147 (1998), 231-241.
|
[13] |
I. Chueshov, M. Eller and I. Lasiecka,
On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.
|
[14] |
V. Georgiev and G. Todorova,
Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.
|
[15] |
J.-M. Ghidaglia and A. Marzocchi,
Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879-895.
|
[16] |
R. T. Glassey,
Blow up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203.
|
[17] |
Y. Guo and M. A. Rammaha,
Blow-up of solutions to systems of nonlinear wave equations
with supercritical sources, Appl. Anal., 92 (2013), 1101-1115.
|
[18] |
Y. Guo and M. A. Rammaha,
Global existence and decay of energy to systems of wave equations with damping and supercritical sources, Z. Angew. Math. Phys., 64 (2013), 621-658.
|
[19] |
Y. Guo and M. A. Rammaha,
Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc., 366 (2014), 2265-2325.
|
[20] |
Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. S. Titi and D. Toundykov,
Hadamard wellposedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812.
|
[21] |
H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, In Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), volume 50 of Progr. Nonlinear Differential Equations Appl., pages 197-216. Birkhäuser, Basel, 2002. |
[22] |
H. A. Levine,
Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = −Au + $ \mathcal{F} $(u), Trans. Amer. Math. Soc., 192 (1974), 1-21.
|
[23] |
J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York, 1972. |
[24] |
J.-L. Lions and W. A. Strauss,
Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.
|
[25] |
M. Nakao and T. Nanbu,
Existence of global (bounded) solutions for some nonlinear evolution equations of second order, Math. Rep. College General Ed. Kyushu Univ., 10 (1975), 67-75.
|
[26] |
P. Pei, M. A. Rammaha and D. Toundykov, Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources, J. Math. Phys., 56 (2015), 081503, 30. |
[27] |
P. Radu,
Weak solutions to the initial boundary value problem for a semilinear wave equation with damping and source terms, Appl. Math. (Warsaw), 35 (2008), 355-378.
|
[28] |
M. Rammaha, D. Toundykov and Z. Wilstein,
Global existence and decay of energy for a nonlinear wave equation with p-Laplacian damping, Discrete Contin. Dyn. Syst., 32 (2012), 4361-4390.
|
[29] |
M. A. Rammaha and Z. Wilstein,
Hadamard well-posedness for wave equations with pLaplacian damping and supercritical sources, Adv. Differential Equations, 17 (2012), 105-150.
|
[30] |
E. Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, Rend. Istit. Mat. Univ. Trieste, 31 (2000), 245-275. Workshop on Blow-up and Global Existence of Solutions for Parabolic and Hyperbolic Problems (Trieste, 1999). |
[31] |
E. Vitillaro,
Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298.
|
[32] |
E. Vitillaro,
A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395.
|
[33] |
G. F. Webb,
Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32 (1980), 631-643.
|
show all references
References:
[1] |
R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975. |
[2] |
K. Agre and M. A. Rammaha,
Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations, 19 (2006), 1235-1270.
|
[3] |
P. Aviles and J. Sandefur,
Nonlinear second order equations with applications to partial differential equations, J. Differential Equations, 58 (1985), 404-427.
|
[4] |
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, 2010. |
[5] |
A. Benaissa and S. Mokeddem,
Decay estimates for the wave equation of p-Laplacian type with dissipation of m-Laplacian type, Math. Methods Appl. Sci., 30 (2007), 237-247.
|
[6] |
A. C. Biazutti,
On a nonlinear evolution equation and its applications, Nonlinear Anal., 24 (1995), 1221-1234.
|
[7] |
L. Bociu and I. Lasiecka,
Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Appl. Math. (Warsaw), 35 (2008), 281-304.
|
[8] |
L. Bociu and I. Lasiecka,
Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860.
|
[9] |
L. Bociu and I. Lasiecka,
Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.
|
[10] |
F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible NAvierSTokes Equations and Related Models, Springer, 2013. |
[11] |
M. M. Cavalcanti and V. N. Domingos,
Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping-source interaction, J. Differential Equations, 236 (2007), 407-459.
|
[12] |
F. Chen, B. Guo and P. Wang,
Long time behavior of strongly damped nonlinear wave equations, J. Differential Equations, 147 (1998), 231-241.
|
[13] |
I. Chueshov, M. Eller and I. Lasiecka,
On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.
|
[14] |
V. Georgiev and G. Todorova,
Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.
|
[15] |
J.-M. Ghidaglia and A. Marzocchi,
Longtime behaviour of strongly damped wave equations, global attractors and their dimension, SIAM J. Math. Anal., 22 (1991), 879-895.
|
[16] |
R. T. Glassey,
Blow up theorems for nonlinear wave equations, Math. Z., 132 (1973), 183-203.
|
[17] |
Y. Guo and M. A. Rammaha,
Blow-up of solutions to systems of nonlinear wave equations
with supercritical sources, Appl. Anal., 92 (2013), 1101-1115.
|
[18] |
Y. Guo and M. A. Rammaha,
Global existence and decay of energy to systems of wave equations with damping and supercritical sources, Z. Angew. Math. Phys., 64 (2013), 621-658.
|
[19] |
Y. Guo and M. A. Rammaha,
Systems of nonlinear wave equations with damping and supercritical boundary and interior sources, Trans. Amer. Math. Soc., 366 (2014), 2265-2325.
|
[20] |
Y. Guo, M. A. Rammaha, S. Sakuntasathien, E. S. Titi and D. Toundykov,
Hadamard wellposedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812.
|
[21] |
H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, In Evolution Equations, Semigroups and Functional Analysis (Milano, 2000), volume 50 of Progr. Nonlinear Differential Equations Appl., pages 197-216. Birkhäuser, Basel, 2002. |
[22] |
H. A. Levine,
Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt = −Au + $ \mathcal{F} $(u), Trans. Amer. Math. Soc., 192 (1974), 1-21.
|
[23] |
J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York, 1972. |
[24] |
J.-L. Lions and W. A. Strauss,
Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.
|
[25] |
M. Nakao and T. Nanbu,
Existence of global (bounded) solutions for some nonlinear evolution equations of second order, Math. Rep. College General Ed. Kyushu Univ., 10 (1975), 67-75.
|
[26] |
P. Pei, M. A. Rammaha and D. Toundykov, Weak solutions and blow-up for wave equations of p-Laplacian type with supercritical sources, J. Math. Phys., 56 (2015), 081503, 30. |
[27] |
P. Radu,
Weak solutions to the initial boundary value problem for a semilinear wave equation with damping and source terms, Appl. Math. (Warsaw), 35 (2008), 355-378.
|
[28] |
M. Rammaha, D. Toundykov and Z. Wilstein,
Global existence and decay of energy for a nonlinear wave equation with p-Laplacian damping, Discrete Contin. Dyn. Syst., 32 (2012), 4361-4390.
|
[29] |
M. A. Rammaha and Z. Wilstein,
Hadamard well-posedness for wave equations with pLaplacian damping and supercritical sources, Adv. Differential Equations, 17 (2012), 105-150.
|
[30] |
E. Vitillaro, Some new results on global nonexistence and blow-up for evolution problems with positive initial energy, Rend. Istit. Mat. Univ. Trieste, 31 (2000), 245-275. Workshop on Blow-up and Global Existence of Solutions for Parabolic and Hyperbolic Problems (Trieste, 1999). |
[31] |
E. Vitillaro,
Global existence for the wave equation with nonlinear boundary damping and source terms, J. Differential Equations, 186 (2002), 259-298.
|
[32] |
E. Vitillaro,
A potential well theory for the wave equation with nonlinear source and boundary damping terms, Glasg. Math. J., 44 (2002), 375-395.
|
[33] |
G. F. Webb,
Existence and asymptotic behavior for a strongly damped nonlinear wave equation, Canad. J. Math., 32 (1980), 631-643.
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