The two-dimensional Keller–Segel system with singular sensitivity and signal absorption: Global large-data solutions and their relaxation properties

2016 ◽  
Vol 26 (05) ◽  
pp. 987-1024 ◽  
Author(s):  
Michael Winkler
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
A Priori Estimates ◽  
Large Data ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Existence Theory ◽  
Relaxation Properties ◽  
Global Existence Of Solutions

We consider the chemotaxis system [Formula: see text] as originally introduced in 1971 by Keller and Segel in the second of their seminal works. This system constitutes a prototypical model for taxis-driven pattern formation and front propagation in various biological contexts such as tumor angiogenesis, but in the higher-dimensional context any global existence theory for large-data solutions is yet lacking. In this work it is shown that in bounded planar domains [Formula: see text] with smooth boundary, for all reasonably regular initial data [Formula: see text] and [Formula: see text], the corresponding Neumann initial-boundary value problem possesses a global generalized solution. Thus particularly addressing arbitrarily large initial data, this goes beyond previously gained results asserting global existence of solutions only in spatial one-dimensional problems, or under certain smallness conditions on the initial data. The derivation of this result is based on a priori estimates for the quantities [Formula: see text] and [Formula: see text] in spatio-temporal [Formula: see text] spaces, where further boundedness and compactness properties are derived from the former by relying on the planar spatial setting in using an associated Moser–Trudinger inequality. Furthermore, some further boundedness and relaxation properties are derived, inter alia indicating that for any such solution we have [Formula: see text] in [Formula: see text] as [Formula: see text] for all finite [Formula: see text], and that in an appropriate generalized sense the quantities [Formula: see text] and [Formula: see text] eventually enter bounded sets in [Formula: see text] and [Formula: see text], respectively, with diameters only determined by the total population size [Formula: see text]. Finally, some numerical experiments illustrate the analytically obtained results.

Global Existence of Solutions for some Nonlinear Wave Equation

2014 ◽  
Vol 638-640 ◽  
pp. 1700-1704
Author(s):  
Yue Hu
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Nonlinear Wave Equation ◽  
Existence Of Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Dissipative Term ◽  
Global Existence Of Solutions ◽  
Initial Boundary Value

In this paper, we consider the existence of global solution to the initial-boundary value problem for some hyperbolic equation with P-Laplace operator and a nonlinear dissipative term using the compactness criteria and the monotone mapping’s method.

FRACTIONAL LANDAU–GINZBURG EQUATIONS ON A SEGMENT

2008 ◽  
Vol 10 (06) ◽  
pp. 1151-1181
Author(s):  
ELENA I. KAIKINA
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Existence Of Solutions ◽  
Asymptotic Representation Of Solutions ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We study the initial-boundary value problem for the fractional Landau–Ginzburg equations on a segment. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem and to find the main term of the asymptotic representation of solutions.

scholarly journals On an Initial Boundary Value Problem for a Class of Odd Higher Order Pseudohyperbolic Integrodifferential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12
Author(s):  
Said Mesloub
Keyword(s):  
Boundary Value Problem ◽  
A Priori Estimates ◽  
Boundary Value ◽  
Nonlocal Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Higher Order ◽  
Well Posedness ◽  
Initial Boundary Value

This paper is devoted to the study of the well-posedness of an initial boundary value problem for an odd higher order nonlinear pseudohyperbolic integrodifferential partial differential equation. We associate to the equationnnonlocal conditions andn+1classical conditions. Upon some a priori estimates and density arguments, we first establish the existence and uniqueness of the strongly generalized solution in a class of a certain type of Sobolev spaces for the associated linear mixed problem. On the basis of the obtained results for the linear problem, we apply an iterative process in order to establish the well-posedness of the nonlinear problem.

scholarly journals Critical convective-type equations on a half-line

2006 ◽  
Vol 2006 ◽  
pp. 1-24
Author(s):  
Elena I. Kaikina
Keyword(s):  
Global Existence ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Time Behavior ◽  
Global Existence Of Solutions ◽  
Representation Of Solutions ◽  
Initial Boundary Value

We are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for critical convective-type dissipative equationsut+ℕ(u,ux)+(an∂xn+am∂xm)u=0,(x,t)∈ℝ+×ℝ+,u(x,0)=u0(x),x∈ℝ+,∂xj−1u(0,t)=0forj=1,…,m/2, where the constantsan,am∈ℝ,n,mare integers, the nonlinear termℕ(u,ux)depends on the unknown functionuand its derivativeuxand satisfies the estimate|ℕ(u,v)|≤C|u|ρ|v|σwithσ≥0,ρ≥1, such that((n+2)/2n)(σ+ρ−1)=1,ρ≥1,σ∈[0,m). Also we suppose that∫ℝ+xn/2ℕdx=0. The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem above-mentioned. We find the main term of the asymptotic representation of solutions in critical case. Also we give some general approach to obtain global existence of solution of initial-boundary value problem in critical convective case and elaborate general sufficient conditions to obtain asymptotic expansion of solution.

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>

scholarly journals Global existence and nonlinear diffusion of classical solutions to non-isentropic Euler equations with damping in bounded domain

2016 ◽  
Vol 13 (01) ◽  
pp. 147-179
Author(s):  
Fuzhou Wu
Keyword(s):  
Global Existence ◽  
Euler Equations ◽  
Nonlinear Diffusion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Existence Theory ◽  
Classical Solutions ◽  
Nonlinear Diffusion Equations ◽  
Isentropic Euler Equations ◽  
Initial Boundary Value

We consider classical solutions to the initial boundary value problem for the non-isentropic compressible Euler equations with damping in several space dimensions. We establish a global existence theory for classical solutions to this system when the initial data are sufficiently small. We prove that the pressure and the velocity converge exponentially toward some constants, while the entropy and the density in general do not approach constants. Finally, we prove that, as the time goes to infinity, the pressure and the velocity components of the non-isentropic Euler equations with damping converge exponentially toward those of an associated nonlinear diffusion system, and the global existence of classical solutions to these nonlinear diffusion equations is also established.

scholarly journals Finite difference approximation of a parabolic problem with variable coefficients

2014 ◽  
Vol 95 (109) ◽  
pp. 49-62 ◽  
Author(s):  
Bosko Jovanovic ◽  
Zorica Milovanovic
Keyword(s):  
Finite Difference ◽  
A Priori Estimates ◽  
A Priori ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Generalized Solution ◽  
Variable Coefficients ◽  
Finite Difference Approximation ◽  
Difference Approximation ◽  
Convergence Rate Estimate

We study the convergence of a finite difference scheme that approximates the third initial-boundary-value problem for a parabolic equation with variable coefficients on a unit square. We assume that the generalized solution of the problem belongs to the Sobolev space W s,s/2 2, s?3. An almost second-order convergence rate estimate (with additional logarithmic factor) in the discrete W 1,1/2 2 norm is obtained. The result is based on some nonstandard a priori estimates involving fractional order discrete Sobolev norms.

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