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

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  • 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.

    Mathematics Subject Classification: 2000 Mathematics Subject Classification: 34C11, 34D05, 34D30, 34G20, 35B40.

    Citation:

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  • [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. CherguiConvergence 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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