Locally Lipschitz perturbations of bisemigroups

Pages: 327 - 349, Volume 9, Issue 2, March 2010

doi:10.3934/cpaa.2010.9.327       Abstract        Full Text (290.3K)       Related Articles

Mohamed Sami ElBialy - Department of Mathematics, University of Toledo, Toledo, Ohio 43606, United States (email)

Abstract: 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 $y\in \Omega$.

Keywords:  Bi-semigroups, solitary waves, modulated waves, elliptic equations, exponential dichotomies, evolution equations, block operators, Semigroup perturbations, Riccati equations, ill posed problems.
Mathematics Subject Classification:  Primary: 47D03, 47D06, 34D09; Secondary: 74J35, 35J60, 37L05.

Received: March 2008;      Revised: April 2009;      Published: December 2009.