# American Institute of Mathematical Sciences

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

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

Received  September 2010 Revised  December 2010 Published  March 2011

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.

Citation: Alain Haraux. On the fast solution of evolution equations with a rapidly decaying source term. Mathematical Control & 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 (1973).   Google Scholar [2] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptotic Analysis, 69 (2010), 31.   Google Scholar [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.  Google Scholar [4] R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2000), 1017.  doi: doi:10.1016/S0362-546X(03)00037-3.  Google Scholar [5] A. Haraux, Slow and fast decay of solutions to some second order evolution equations,, J. Anal. Math., 95 (2005), 297.  doi: doi:10.1007/BF02791505.  Google Scholar [6] A. Haraux, Sharp decay estimates of the solutions to a class of nonlinear second order ODE,, Analysis and Applications, 9 (2011), 49.  doi: doi:10.1142/S021953051100173X.  Google Scholar [7] A. Haraux, "Semi-linear Hyperbolic Problems in Bounded Domains," Mathematical reports Vol. 3, Part 1, J. Dieudonn Editor,, Harwood Academic Publishers, (1987).   Google Scholar [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.  doi: doi:10.1504/IJMMNO.2009.030093.  Google Scholar [9] S. Z. Huang and P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, Nonlinear Anal., 46 (2001), 675.  doi: doi:10.1016/S0362-546X(00)00145-0.  Google Scholar [10] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).   Google Scholar

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##### References:
 [1] H. Brézis, "Opérateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert,", North-Holland Mathematics Studies, 5 (1973).   Google Scholar [2] I. Ben Hassen, Decay estimates to equilibrium for some asymptotically autonomous semilinear evolution equations,, Asymptotic Analysis, 69 (2010), 31.   Google Scholar [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.  Google Scholar [4] R. Chill and M. A. Jendoubi, Convergence to steady states in asymptotically autonomous semilinear evolution equations,, Nonlinear Anal., 53 (2000), 1017.  doi: doi:10.1016/S0362-546X(03)00037-3.  Google Scholar [5] A. Haraux, Slow and fast decay of solutions to some second order evolution equations,, J. Anal. Math., 95 (2005), 297.  doi: doi:10.1007/BF02791505.  Google Scholar [6] A. Haraux, Sharp decay estimates of the solutions to a class of nonlinear second order ODE,, Analysis and Applications, 9 (2011), 49.  doi: doi:10.1142/S021953051100173X.  Google Scholar [7] A. Haraux, "Semi-linear Hyperbolic Problems in Bounded Domains," Mathematical reports Vol. 3, Part 1, J. Dieudonn Editor,, Harwood Academic Publishers, (1987).   Google Scholar [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.  doi: doi:10.1504/IJMMNO.2009.030093.  Google Scholar [9] S. Z. Huang and P. Takac, Convergence in gradient-like systems which are asymptotically autonomous and analytic,, Nonlinear Anal., 46 (2001), 675.  doi: doi:10.1016/S0362-546X(00)00145-0.  Google Scholar [10] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,", Applied Mathematical Sciences, 44 (1983).   Google Scholar
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