On the fast solution of evolution equations with a rapidly decaying source term

Previous Article
Numerical methods for dividend optimization using regime-switching jump-diffusion models
MCRF Home
This Issue
Next Article

March 2011, 1(1): 1-20. doi: 10.3934/mcrf.2011.1.1

On the fast solution of evolution equations with a rapidly decaying source term

Alain Haraux ^1,

1.	UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received September 2010 Revised December 2010 Published March 2011

Abstract
Related Papers

If $L$ is the generator of a uniformly bounded group of operators $T(t)$ on a Banach space $X$, the abstract evolution equation $ u' + Lu(t) = h(t) $ has a (weak) solution tending to $0$ as $t\rightarrow +\infty $ if, and only if $\int_0^{+\infty}T(s) h(s) ds $ is semi-convergent, and then this solution is unique. For the semi-linear equation $ u' + Lu(t) + f(u) = h(t) $, if $f$ such that $f(0) = 0$ is Lipschitz continuous on bounded subsets of $X$ and has a Lipschitz constant bounded by $ Cr^\alpha $ in the ball $B(0, r)$ for $r\leq r_0$, for any $h$ satisfiying

$||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.

Keywords: rapidly decaying source., Fast solution, evolution equations.

Mathematics Subject Classification: 2000 Mathematics Subject Classification: 34C11, 34D05, 34D30, 34G20, 35B4.

Citation: Alain Haraux. On the fast solution of evolution equations with a rapidly decaying source term. Mathematical Control and Related Fields, 2011, 1 (1) : 1-20. doi: 10.3934/mcrf.2011.1.1

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.

[1]	Xiaoming An, Xian Yang. Semi-classical states for fractional Schrödinger equations with magnetic fields and fast decaying potentials. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1649-1672. doi: 10.3934/cpaa.2022038
[2]	Shoshana Kamin, Guillermo Reyes, Juan Luis Vázquez. Long time behavior for the inhomogeneous PME in a medium with rapidly decaying density. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 521-549. doi: 10.3934/dcds.2010.26.521
[3]	Kazuhiro Ishige, Y. Kabeya. Hot spots for the two dimensional heat equation with a rapidly decaying negative potential. Discrete and Continuous Dynamical Systems - S, 2011, 4 (4) : 833-849. doi: 10.3934/dcdss.2011.4.833
[4]	D. Wirosoetisno. Navier--Stokes equations on a rapidly rotating sphere. Discrete and Continuous Dynamical Systems - B, 2015, 20 (4) : 1251-1259. doi: 10.3934/dcdsb.2015.20.1251
[5]	Vitalii G. Kurbatov, Valentina I. Kuznetsova. On stability of functional differential equations with rapidly oscillating coefficients. Communications on Pure and Applied Analysis, 2018, 17 (1) : 267-283. doi: 10.3934/cpaa.2018016
[6]	Walter Allegretto, Liqun Cao, Yanping Lin. Multiscale asymptotic expansion for second order parabolic equations with rapidly oscillating coefficients. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 543-576. doi: 10.3934/dcds.2008.20.543
[7]	Liping Wang, Chunyi Zhao. Infinitely many solutions for nonlinear Schrödinger equations with slow decaying of potential. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1707-1731. doi: 10.3934/dcds.2017071
[8]	Zuji Guo. Nodal solutions for nonlinear Schrödinger equations with decaying potential. Communications on Pure and Applied Analysis, 2016, 15 (4) : 1125-1138. doi: 10.3934/cpaa.2016.15.1125
[9]	François Genoud, Charles A. Stuart. Schrödinger equations with a spatially decaying nonlinearity: Existence and stability of standing waves. Discrete and Continuous Dynamical Systems, 2008, 21 (1) : 137-186. doi: 10.3934/dcds.2008.21.137
[10]	Alexandre Caboussat, Allison Leonard. Numerical solution and fast-slow decomposition of a population of weakly coupled systems. Conference Publications, 2009, 2009 (Special) : 123-132. doi: 10.3934/proc.2009.2009.123
[11]	Kin Ming Hui, Jinwan Park. Asymptotic behaviour of singular solution of the fast diffusion equation in the punctured euclidean space. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5473-5508. doi: 10.3934/dcds.2021085
[12]	Jie Zhao. Large time behavior of solution to quasilinear chemotaxis system with logistic source. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1737-1755. doi: 10.3934/dcds.2020091
[13]	Langhao Zhou, Liangwei Wang, Chunhua Jin. Global solvability to a singular chemotaxis-consumption model with fast and slow diffusion and logistic source. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2065-2075. doi: 10.3934/dcdsb.2021122
[14]	José M. Arrieta, Simone M. Bruschi. Very rapidly varying boundaries in equations with nonlinear boundary conditions. The case of a non uniformly Lipschitz deformation. Discrete and Continuous Dynamical Systems - B, 2010, 14 (2) : 327-351. doi: 10.3934/dcdsb.2010.14.327
[15]	Yun-Gang Chen, Yoshikazu Giga, Koh Sato. On instant extinction for very fast diffusion equations. Discrete and Continuous Dynamical Systems, 1997, 3 (2) : 243-250. doi: 10.3934/dcds.1997.3.243
[16]	Zhongying Chen, Bin Wu, Yuesheng Xu. Fast numerical collocation solutions of integral equations. Communications on Pure and Applied Analysis, 2007, 6 (3) : 643-666. doi: 10.3934/cpaa.2007.6.643
[17]	Alexandre Nolasco de Carvalho, Jan W. Cholewa, Tomasz Dlotko. Damped wave equations with fast growing dissipative nonlinearities. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1147-1165. doi: 10.3934/dcds.2009.24.1147
[18]	Yalçin Sarol, Frederi Viens. Time regularity of the evolution solution to fractional stochastic heat equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 895-910. doi: 10.3934/dcdsb.2006.6.895
[19]	Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022
[20]	Md. Abul Kalam Azad, Edite M.G.P. Fernandes. A modified differential evolution based solution technique for economic dispatch problems. Journal of Industrial and Management Optimization, 2012, 8 (4) : 1017-1038. doi: 10.3934/jimo.2012.8.1017

2020 Impact Factor: 1.284

Tools

Metrics

Other articles
by authors

on AIMS
Alain Haraux
on Google Scholar
Alain Haraux

[Back to Top]