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March  2010, 9(2): 327-349. doi: 10.3934/cpaa.2010.9.327

Locally Lipschitz perturbations of bisemigroups


Department of Mathematics, University of Toledo, Toledo, Ohio 43606, United States

Received  March 2008 Revised  April 2009 Published  December 2009

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 $y\in \Omega$.

Citation: Mohamed Sami ElBialy. Locally Lipschitz perturbations of bisemigroups. Communications on Pure and Applied Analysis, 2010, 9 (2) : 327-349. doi: 10.3934/cpaa.2010.9.327

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