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On the fast solution of evolution equations with a rapidly decaying source term
1. | UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France |
$||h(t)|| \leq c(1+t)^{-(1+ \lambda )} $
with $\lambda >\frac{1}{\alpha}$ and $c$ small enough there exists a unique solution tending to $0$ at least like $(1+t)^{- \lambda}.$ When the system is dissipative, this special solution makes it sometimes possible to estimate from below the rate of decay to $0$ of the other solutions.
References:
[1] |
H. Brézis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert," North-Holland Mathematics Studies, 5, Amsterdam-London, New York, 1973. |
[2] |
I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, Asymptotic Analysis, 69 (2010), 31-44. |
[3] |
I. Ben Hassen and L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation,, Journal of Dynamics and Differential Equations., ().
doi: 10.1007/s10884-011-9212-7. |
[4] |
R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2000), 1017-1039.
doi: doi:10.1016/S0362-546X(03)00037-3. |
[5] |
A. Haraux, Slow and fast decay of solutions to some second order evolution equations, J. Anal. Math., 95 (2005), 297-321.
doi: doi:10.1007/BF02791505. |
[6] |
A. Haraux, Sharp decay estimates of the solutions to a class of nonlinear second order ODE, Analysis and Applications, 9 (2011), 49-69.
doi: doi:10.1142/S021953051100173X. |
[7] |
A. Haraux, "Semi-linear Hyperbolic Problems in Bounded Domains," Mathematical reports Vol. 3, Part 1, J. Dieudonn Editor, Harwood Academic Publishers, Gordon & Breach, 1987. |
[8] |
A. Haraux, $L^p$ estimates of solutions to some nonlinear wave equations in one space dimension, International Journal of Mathematical Modelling and Numerical Optimisation, 1 (2009), 146-152.
doi: doi:10.1504/IJMMNO.2009.030093. |
[9] |
S. Z. Huang and P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., 46 (2001), 675-698.
doi: doi:10.1016/S0362-546X(00)00145-0. |
[10] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. |
show all references
References:
[1] |
H. Brézis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert," North-Holland Mathematics Studies, 5, Amsterdam-London, New York, 1973. |
[2] |
I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations, Asymptotic Analysis, 69 (2010), 31-44. |
[3] |
I. Ben Hassen and L. Chergui, Convergence of global and bounded solutions of some nonautonomous second order evolution equations with nonlinear dissipation,, Journal of Dynamics and Differential Equations., ().
doi: 10.1007/s10884-011-9212-7. |
[4] |
R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations, Nonlinear Anal., 53 (2000), 1017-1039.
doi: doi:10.1016/S0362-546X(03)00037-3. |
[5] |
A. Haraux, Slow and fast decay of solutions to some second order evolution equations, J. Anal. Math., 95 (2005), 297-321.
doi: doi:10.1007/BF02791505. |
[6] |
A. Haraux, Sharp decay estimates of the solutions to a class of nonlinear second order ODE, Analysis and Applications, 9 (2011), 49-69.
doi: doi:10.1142/S021953051100173X. |
[7] |
A. Haraux, "Semi-linear Hyperbolic Problems in Bounded Domains," Mathematical reports Vol. 3, Part 1, J. Dieudonn Editor, Harwood Academic Publishers, Gordon & Breach, 1987. |
[8] |
A. Haraux, $L^p$ estimates of solutions to some nonlinear wave equations in one space dimension, International Journal of Mathematical Modelling and Numerical Optimisation, 1 (2009), 146-152.
doi: doi:10.1504/IJMMNO.2009.030093. |
[9] |
S. Z. Huang and P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic, Nonlinear Anal., 46 (2001), 675-698.
doi: doi:10.1016/S0362-546X(00)00145-0. |
[10] |
A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations," Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. |
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