scholarly journals Global generalized solutions of a haptotaxis model describing cancer cells invasion and metastatic spread

2012 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Meng Liu ◽  
Yuxiang Li
Keyword(s):  
Initial Data ◽  
Cancer Cells ◽  
Smooth Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Metastatic Spread ◽  
Generalized Solutions ◽  
Neumann Type ◽  
Initial Boundary Value

<p style='text-indent:20px;'>In this paper, we consider the following haptotaxis model describing cancer cells invasion and metastatic spread</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1a"> \begin{document}$\begin{array}{*{20}{c}}{\left\{ {\begin{array}{*{20}{l}}{{u_t} = \Delta u - \chi \nabla \cdot (u\nabla w),}&amp;{x \in \Omega ,\;t &gt; 0,}\\{{v_t} = {d_v}\Delta v - \xi \nabla \cdot (v\nabla w),}&amp;{x \in \Omega ,\;t &gt; 0,}\\{{m_t} = {d_m}\Delta m + u - m,}&amp;{x \in \Omega ,\;t &gt; 0,}\\{{w_t} = - \left( {{\gamma _1}u + m} \right)w,}&amp;{x \in \Omega ,\;t &gt; 0,}\end{array}} \right.}&amp;{(0.1)}\end{array}$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^3 $\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary and the parameters <inline-formula><tex-math id="M2">\begin{document}$ \chi, \xi, d_{v}, d_{m},\gamma_{1}&gt;0 $\end{document}</tex-math></inline-formula>. Under homogeneous boundary conditions of Neumann type for <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ v $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M5">\begin{document}$ m $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M6">\begin{document}$ w $\end{document}</tex-math></inline-formula>, it is proved that, for suitable smooth initial data <inline-formula><tex-math id="M7">\begin{document}$ (u_0, v_0, m_0, w_0) $\end{document}</tex-math></inline-formula>, the corresponding Neumann initial-boundary value problem possesses a global generalized solution.</p>

Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions

2019 ◽  
Vol 29 (03) ◽  
pp. 373-418 ◽  
Author(s):  
Michael Winkler
Keyword(s):  
Nutrient Concentration ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solutions ◽  
Cross Diffusion ◽  
Temporal Decay ◽  
Mild Assumption ◽  
Neumann Type ◽  
Spatio Temporal ◽  
Initial Boundary Value

This work is concerned with a prototypical model for the spatio-temporal evolution of a forager–exploiter system, consisting of two species which simultaneously consume a common nutrient, and which interact through a taxis-type mechanism according to which individuals from the exploiter subpopulation move upward density gradients of the forager subgroup. Specifically, the model [Formula: see text] for the population densities [Formula: see text] and [Formula: see text] of foragers and exploiters, as well as the nutrient concentration [Formula: see text], is considered in smoothly bounded domains [Formula: see text], [Formula: see text]. It is first shown that under an explicit condition linking the sizes of the resource production rate [Formula: see text] and of the initial nutrient concentration, an associated Neumann-type initial-boundary value problem admits a global solution within an appropriate generalized concept. The second of the main results asserts stabilization of these solutions toward spatially homogeneous equilibria in the large time limit, provided that [Formula: see text] satisfies a mild assumption on temporal decay. To the best of our knowledge, these are the first rigorous analytical results addressing taxis-type cross-diffusion mechanisms coupled in a cascade-like manner as in (⋆).

scholarly journals GENERALIZED SOLUTIONS FOR THE EULER–BERNOULLI MODEL WITH ZENER VISCOELASTIC FOUNDATIONS AND DISTRIBUTIONAL FORCES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350017 ◽  
Author(s):  
GÜNTHER HÖRMANN ◽  
SANJA KONJIK ◽  
LJUBICA OPARNICA
Keyword(s):  
Differential Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variational Problems ◽  
Generalized Solutions ◽  
Beam Model ◽  
Order Partial Differential Equation ◽  
Viscoelastic Foundation ◽  
Regularization Techniques ◽  
Initial Boundary Value

We study the initial-boundary value problem for an Euler–Bernoulli beam model with discontinuous bending stiffness laying on a viscoelastic foundation and subjected to an axial force and an external load both of Dirac-type. The corresponding model equation is a fourth-order partial differential equation and involves discontinuous and distributional coefficients as well as a distributional right-hand side. Moreover the viscoelastic foundation is of Zener-type and described by a fractional differential equation with respect to time. We show how functional analytic methods for abstract variational problems can be applied in combination with regularization techniques to prove existence and uniqueness of generalized solutions.

scholarly journals On initial boundary value problem with Dirichlet integral conditions for a hyperbolic equation with the Bessel operator

2003 ◽  
Vol 2003 (10) ◽  
pp. 487-502
Author(s):  
Abdelfatah Bouziani
Keyword(s):  
Hyperbolic Equation ◽  
A Priori ◽  
Weighted Sobolev Spaces ◽  
A Priori Estimate ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Bessel Operator ◽  
Integral Conditions ◽  
Initial Boundary Value

We consider a mixed problem with Dirichlet and integral conditions for a second-order hyperbolic equation with the Bessel operator. The existence, uniqueness, and continuous dependence of a strongly generalized solution are proved. The proof is based on an a priori estimate established in weighted Sobolev spaces and on the density of the range of the operator corresponding to the abstract formulation of the considered problem.

scholarly journals Global well-posedness in a chemotaxis system with oxygen consumption

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Xujie Yang
Keyword(s):  
Smooth Boundary ◽  
Neumann Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Liquid Environment ◽  
Bounded Convex Domain ◽  
Initial Boundary Value ◽  
Chemotaxis System ◽  
Bounded Convex ◽  
Uniformly Bounded

<p style='text-indent:20px;'>Motivated by the studies of the hydrodynamics of the tethered bacteria <i>Thiovulum majus</i> in a liquid environment, we consider the following chemotaxis system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \left\{ \begin{split} &amp; n_t = \Delta n-\nabla\cdot\left(n\chi(c)\nabla{c}\right)+nc, &amp;x\in \Omega, t&gt;0, \ &amp; c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &amp;x\in \Omega, t&gt;0\ \end{split} \right. \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a bounded convex domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^d(d\in\{2, 3\}) $\end{document}</tex-math></inline-formula> with smooth boundary. For any given fluid <inline-formula><tex-math id="M2">\begin{document}$ {\bf u} $\end{document}</tex-math></inline-formula>, it is proved that if <inline-formula><tex-math id="M3">\begin{document}$ d = 2 $\end{document}</tex-math></inline-formula>, the corresponding initial-boundary value problem admits a unique global classical solution which is uniformly bounded, while if <inline-formula><tex-math id="M4">\begin{document}$ d = 3 $\end{document}</tex-math></inline-formula>, such solution still exists under the additional condition that <inline-formula><tex-math id="M5">\begin{document}$ 0&lt;\chi\leq \frac{1}{16\|c(\cdot, 0)\|_{L^\infty(\Omega)}} $\end{document}</tex-math></inline-formula>.</p>

scholarly journals A Nonhomogeneous Boundary-Valued Problem for the coupled KDV system

2021 ◽  
Author(s):  
Yitong Pei ◽  
Boling Guo
Keyword(s):  
Boundary Conditions ◽  
Contraction Mapping ◽  
Finite Interval ◽  
Smooth Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Contraction Mapping Principle ◽  
Nonhomogeneous Boundary ◽  
The Laplace Transform ◽  
Initial Boundary Value

In this paper, we study the initial-boundary-value problem (IBVP) for coupled Korteweg-de Vries equations posed on a finite interval with nonhomogeneous boundary conditions. We overcome the requirement for stronger smooth boundary conditions in the traditional method via the Laplace transform. Our approach uses the strong Kato smoothing property and the contraction mapping principle.

scholarly journals On continuous dependence for the mixed problem of microstretch bodies

2017 ◽  
Vol 25 (1) ◽  
pp. 131-143
Author(s):  
M. Marin ◽  
I. Abbas ◽  
C. Cârstea
Keyword(s):  
Hilbert Space ◽  
Initial Data ◽  
Continuous Dependence ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Linear Operators ◽  
Evolutionary Equation ◽  
Trans Form ◽  
Semigroups Of Linear Operators ◽  
Initial Boundary Value

AbstractWe do a qualitative study on the mixed initial-boundary value problem in the elastodynamic theory of microstretch bodies. After we trans- form this problem in a temporally evolutionary equation on a Hilbert space, we will use some results from the theory of semigroups of linear operators in order to prove the continuous dependence of the solutions upon initial data and supply terms.

A mathematical model for long waves generated by wavemakers in non-linear dispersive systems

1973 ◽  
Vol 73 (2) ◽  
pp. 391-405 ◽  
Author(s):  
J. L. Bona ◽  
P. J. Bryant
Keyword(s):  
Mathematical Model ◽  
Initial Data ◽  
Water Waves ◽  
Open Channel ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Finite Amplitude ◽  
Mathematical Properties ◽  
Image Position ◽  
Initial Boundary Value

An initial-boundary-value problem for the equationis considered for x, t ≥ 0. This system is a model for long water waves of small but finite amplitude, generated in a uniform open channel by a wavemaker at one end. It is shown that, in contrast to an alternative, more familiar model using the Korteweg–deVries equation, the solution of (a) has good mathematical properties: in particular, the problem is well set in Hadamard's classical sense that solutions corresponding to given initial data exist, are unique, and depend continuously on the specified data.

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