PROC
Remarks on the global existence of weak solutions to quasilinear degenerate Keller-Segel systems
Sachiko Ishida Tomomi Yokota
Conference Publications 2013, 2013(special): 345-354 doi: 10.3934/proc.2013.2013.345
The global existence of weak solutions to quasilinear ``degenerate'' Keller-Segel systems is shown in the recent papers [3], [4]. This paper gives some improvements and supplements of these. More precisely, the differentiability and the smallness of initial data are weakened when the spatial dimension $N$ satisfies $N\geq2$. Moreover, the global existence is established in the case $N=1$ which is unsolved in [4].
keywords: global existence. Quasilinear degenerate Keller-Segel systems
DCDS
Quasi-$m$-accretivity of Schrödinger operators with singular first-order coefficients
Noboru Okazawa Tomomi Yokota
Discrete & Continuous Dynamical Systems - A 2008, 22(4): 1081-1090 doi: 10.3934/dcds.2008.22.1081
The Schrödinger operator $T = (i\nabla +b)^2+a \cdot \nabla + q$ on $\mathbb{R}^N$ is considered for $N \ge 2$. Here $a=(a_{j})$ and $b=(b_{j})$ are real-vector-valued functions on $\mathbb{R}^N$, while $q$ is a complex-scalar-valued function on $\mathbb{R}^N$. Over twenty years ago late Professor Kato proved that the minimal realization $T_{min}$ is essentially quasi-$m$-accretive in $L^2(\mathbb{R}^N)$ if, among others, $(1+|x|)^{-1}a_j \in L^4(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$. In this paper it is shown that under some additional conditions the same conclusion remains true even if $a_j \in L^4_{loc}(\mathbb{R}^N)$.
keywords: Schrödinger operators quasi-$m$-accretive operators.
DCDS-B
Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
Sachiko Ishida Tomomi Yokota
Discrete & Continuous Dynamical Systems - B 2013, 18(10): 2569-2596 doi: 10.3934/dcdsb.2013.18.2569
This paper gives a blow-up result for the quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. It is known that the system has a global solvability in the case where $q < m + \frac{2}{N}$ ($m$ denotes the intensity of diffusion and $q$ denotes the nonlinearity) without any restriction on the size of initial data, and where $q \geq m + \frac{2}{N}$ and the initial data are ``small''. However, there is no result when $q \geq m + \frac{2}{N}$ and the initial data are ``large''. This paper discusses such case and shows that there exist blow-up energy solutions from initial data having large negative energy.
keywords: blow-up. Quasilinear degenerate Keller-Segel systems
PROC
Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains
Noboru Okazawa Tomomi Yokota
Conference Publications 2001, 2001(Special): 280-288 doi: 10.3934/proc.2001.2001.280
Please refer to Full Text.
keywords: subdifferential operators Smoothing effect semigroups of nonlinear operators accretive operators the complex Ginzburg-Landau equation.
DCDS
Subdifferential operator approach to strong wellposedness of the complex Ginzburg-Landau equation
Noboru Okazawa Tomomi Yokota
Discrete & Continuous Dynamical Systems - A 2010, 28(1): 311-341 doi: 10.3934/dcds.2010.28.311
Two theorems concerning strong wellposedness are established for the complex Ginzburg-Landau equation. One of them is concerned with strong $L^{2}$-wellposedness, that is, strong wellposedness for $L^{2}$-initial data. The other deals with $H_{0}^{1}$-initial data as a partial extension. By a technical innovation it becomes possible to prove the convergence of approximate solutions without compactness. This type of convergence is known with accretivity methods when the argument of the complex coefficient is small. The new device yields the generation of a class of non-contraction semigroups even when the argument is large. The results are both obtained as application of abstract theory of semilinear evolution equations with subdifferential operators.
keywords: subdifferential operators strong wellposedness smoothing effect accretive operators semigroups of nonlinear operators. The complex Ginzburg-Landau equation
DCDS
Stabilization in a chemotaxis model for tumor invasion
Kentarou Fujie Akio Ito Michael Winkler Tomomi Yokota
Discrete & Continuous Dynamical Systems - A 2016, 36(1): 151-169 doi: 10.3934/dcds.2016.36.151
This paper deals with the chemotaxis system \[ \begin{cases} u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in \Omega, \ t>0, \\ v_t=\Delta v + wz, \qquad x\in \Omega, \ t>0, \\ w_t=-wz, \qquad x\in \Omega, \ t>0, \\ z_t=\Delta z - z + u, \qquad x\in \Omega, \ t>0, \end{cases} \] in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n \le 3$, that has recently been proposed as a model for tumor invasion in which the role of an active extracellular matrix is accounted for.
    It is shown that for any choice of nonnegative and suitably regular initial data $(u_0,v_0,w_0,z_0)$, a corresponding initial-boundary value problem of Neumann type possesses a global solution which is bounded. Moreover, it is proved that whenever $u_0\not\equiv 0$, these solutions approach a certain spatially homogeneous equilibrium in the sense that as $t\to\infty$,
    $u(x,t)\to \overline{u_0}$ ,    $v(x,t) \to \overline{v_0} + \overline{w_0}$,    $w(x,t) \to 0$    and     $z(x,t) \to \overline{u_0}$,     uniformly with respect to $x\in\Omega$, where $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$, $\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$    and    $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.
keywords: Chemotaxis tumor invasion. asymptotic behavior
EECT
Energy methods for abstract nonlinear Schrödinger equations
Noboru Okazawa Toshiyuki Suzuki Tomomi Yokota
Evolution Equations & Control Theory 2012, 1(2): 337-354 doi: 10.3934/eect.2012.1.337
So far there seems to be no abstract formulations for nonlinear Schrödinger equations (NLS). In some sense Cazenave[2, Chapter 3] has given a guiding principle to replace the free Schrödinger group with the approximate identity of resolvents. In fact, he succeeded in separating the existence theory from the Strichartz estimates. This paper is a proposal to extend his guiding principle by using the square root of the resolvent. More precisely, the abstract theory here unifies the local existence of weak solutions to (NLS) with not only typical nonlinearities but also some critical cases. Moreover, the theory yields the improvement of [21].
keywords: abstract nonlinear Schrödinger equation inverse-square potentials. Energy methods nonnegative selfadjoint operators
PROC
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$
Kentarou Fujie Chihiro Nishiyama Tomomi Yokota
Conference Publications 2015, 2015(special): 464-472 doi: 10.3934/proc.2015.0464
This paper is concerned with global existence and boundedness of classical solutions to the quasilinear fully parabolic Keller-Segel system $u_t = \nabla \cdot(D(u)\nabla u) -\nabla \cdot (v^{-1}S(u)\nabla v)$, $v_t= \Delta v-v+u$. In [7,4], global existence and boundedness were established in the system without $v^{-1}$. In this paper the signal-dependent sensitivity $v^{-1}$ is taken into account via the Weber-Fechner law. A uniform-in-time estimate for $v$ obtained in [2] defeats the singularity of $v^{-1}$.
keywords: Quasilinear Keller-Segel system singular sensitivity. boundedness
DCDS-B
Gradient estimate for solutions to quasilinear non-degenerate Keller-Segel systems on $\mathbb{R}^N$
Sachiko Ishida Yusuke Maeda Tomomi Yokota
Discrete & Continuous Dynamical Systems - B 2013, 18(10): 2537-2568 doi: 10.3934/dcdsb.2013.18.2537
This paper gives the gradient estimate for solutions to the quasilinear non-degenerate parabolic-parabolic Keller-Segel system (KS) on the whole space $\mathbb{R}^N$. The gradient estimate for (KS) on bounded domains is known as an application of Amann's existence theory in [1]. However, in the whole space case it seems necessary to derive the gradient estimate directly. The key to the proof is a modified Bernstein's method. The result is useful to obtain the whole space version of the global existence result by Tao-Winkler [13] except for the boundedness.
keywords: blow-up. Quasilinear non degenerate Keller-Segel systems

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