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Locally Lipschitz perturbations of bisemigroups

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  • In this work we study the ill posed semilinear system $\dot{x}= Lx + f(\xi,t), \dot{y}= Ry + g(\xi,t)$, $\xi=(x,y)$, in Banach spaces where $L$ and $R$ are the infinitesimal generators of two $C_o$ semigroups $\{\mathcal{L}(t), t\geq 0\}$ and $\{\mathcal{R}(-t), t\geq 0\}$ respectively. The nonlinearity $h=(f,g)$ is continuous in $t$ and locally Lipschitz continuous in $\xi$ locally uniformly in $t$.
    $\cdot $ We show the existence and uniqueness of what we call local dichotomous mild solutions (DMS) that take the form

    $x(t) = e^{(t-t_1)L}x_1 + \int_{t_1}^{t} e^{(t-s)L} f(\xi(s), s)ds$

    $ y(t)= e^{-(t_2-t)R} y_2 - \int_{t}^{t_2} e^{-(s-t)R} g(\xi(s), s) ds$

    $ t_1 < t_2, \qquad t_1\leq t \leq t_2$

    for any sufficiently small time interval $[t_1, t_2]$ and any given $\xi :=(x_1, y_2)$ in a sufficiently small neighbourhood.
    $\cdot $ We show that in the uniform $C^0$-norm DMSs vary continuously with $[t_1, t_2]$ and Lipschitz-continuously with $\xi $.
    $\cdot $ We study the regularity of DMSs under various hypotheses.
    $\cdot $ A simple example that leads to a bisemigroup is a semilinear elliptic system that arises when considering solitary waves in an infinite cylinder:

    $u_{x x}+\Delta u = f(u), \quad u|_{\Gamma} = 0, \quad\Gamma= \mathbb{R}\times \partial\Omega, \quad (x, y, u)\in \mathbb{R}\times \Omega\times\mathbb{R}^m

    where $\Omega$ is a bounded region in $ \mathbb{R}^n$ with $C^2$ boundary and $\Delta$ is the Laplacian in the variable $y\in \Omega$.

    Mathematics Subject Classification: Primary: 47D03, 47D06, 34D09; Secondary: 74J35, 35J60, 37L05.


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