### A blow up condition for a nonautonomous semilinear system.

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We consider initial boundary problems of a two-chemical substances chemotaxis system. In the four-dimensional setting, it was shown that solutions exist globally in time and remain bounded if the total mass is less than ${\left(8\pi \right)}^{2}$, whereas the solution emanating from some initial data of large magnitude may blows up. This result can be regarded as a generalization of the well-known $8\pi $ problem in the Keller–Segel system to higher dimensions. We will compare mathematical structures of the Keller–Segel system...

This note contains some remarks on the paper of Y. Naito concerning the parabolic system of chemotaxis and published in this volume.

We propose a modification of the classical Boussinesq approximation for buoyancy-driven flows of viscous, incompressible fluids in situations where viscous heating cannot be neglected. This modification is motivated by unresolved issues regarding the global solvability of the original system. A very simple model problem leads to a coupled system of two parabolic equations with a source term involving the square of the gradient of one of the unknowns. Based on adequate notions of weak and strong...

We consider a phase-field model of grain structure evolution, which appears in materials sciences. In this paper we study the grain boundary motion model of Kobayashi-Warren-Carter type, which contains a singular diffusivity. The main objective of this paper is to show the existence of solutions in a generalized sense. Moreover, we show the uniqueness of solutions for the model in one-dimensional space.

A model of tumor growth in a spatial environment is analyzed. The model includes proliferating and quiescent compartments of tumor cells indexed by successively mutated cell phenotypes of increasingly proliferative aggressiveness. The model incorporates spatial dependence due to both random motility and directed movement haptotaxis. The model structures tumor cells by both cell age and cell size. The model consists of a system of nonlinear partial differential equations for the compartments of...

We study the initial value problem for the drift-diffusion model arising in semiconductor device simulation and plasma physics. We show that the corresponding stationary problem in the whole space ℝn admits a unique stationary solution in a general situation. Moreover, it is proved that when n ≥ 3, a unique solution to the initial value problem exists globally in time and converges to the corresponding stationary solution as time tends to infinity, provided that the amplitude of the stationary solution...

We consider a nonlinear parabolic system modelling chemotaxis ${u}_{t}=\nabla \xb7(\nabla u-u\nabla v)$, ${v}_{t}=\Delta v+u$ in ℝ², t > 0. We first prove the existence of time-global solutions, including self-similar solutions, for small initial data, and then show the asymptotically self-similar behavior for a class of general solutions.

An abstract semilinear parabolic equation in a Banach space X is considered. Under general assumptions on nonlinearity this problem is shown to generate a bounded dissipative semigroup on ${X}^{\alpha}$. This semigroup possesses an $({X}^{\alpha}-Z)$-global attractor that is closed, bounded, invariant in ${X}^{\alpha}$, and attracts bounded subsets of ${X}^{\alpha}$ in a ’weaker’ topology of an auxiliary Banach space Z. The abstract approach is finally applied to the scalar parabolic equation in Rⁿ and to the partly dissipative system.

This paper provides blow up results of Fujita type for a reaction-diffusion system of 3 equations in the form $u\u209c-\Delta \left({a}_{11}u\right)=h(t,x){\left|v\right|}^{p}$, $v\u209c-\Delta \left({a}_{21}u\right)-\Delta \left({a}_{22}v\right)=k(t,x){\left|w\right|}^{q}$, $w\u209c-\Delta \left({a}_{31}u\right)-\Delta \left({a}_{32}v\right)-\Delta \left({a}_{33}w\right)=l(t,x){\left|u\right|}^{r}$, for $x\in {\mathbb{R}}^{N}$, t > 0, p > 0, q > 0, r > 0, ${a}_{ij}={a}_{ij}(t,x,u,v)$, under initial conditions u(0,x) = u₀(x), v(0,x) = v₀(x), w(0,x) = w₀(x) for $x\in {\mathbb{R}}^{N}$, where u₀, v₀, w₀ are nonnegative, continuous and bounded functions. Subject to conditions on dependence on the parameters p, q, r, N and the growth of the functions h, k, l at infinity, we prove finite blow up time for every solution of the above system,...

We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. ${u}_{it}-{d}_{i}\Delta {u}_{i}={\prod}_{k=1}^{m}{u}_{k}^{{p}_{k}^{i}},i=1,...,m,x\in {\mathbb{R}}^{N},t>0,$ with nonnegative, bounded, continuous initial values and ${p}_{k}^{i}\ge 0$, $i,k=1,...,m$, ${d}_{i}>0$, $i=1,...,m$. For solutions which blow up at $t=T<\le \infty $, we derive the following bounds on the blow up rate: ${u}_{i}(x,t)\le C{(T-t)}^{-{\alpha}_{i}}$ with C > 0 and ${\alpha}_{i}$ defined in terms of ${p}_{k}^{i}$.

A system of one-dimensional linear parabolic equations coupled by boundary conditions which include additional state variables, is considered. This system describes an electric circuit with distributed parameter lines and lumped capacitors all connected through a resistive multiport. By using the monotony in a space of the form ${L}^{2}(0,T;{H}^{1})$, one proves the existence and uniqueness of a variational solution, if reasonable engineering hypotheses are fulfilled.

A class of quasilinear parabolic systems with quadratic nonlinearities in the gradient is considered. It is assumed that the elliptic operator of a system has variational structure. In the multidimensional case, the behavior of solutions of the Cauchy-Dirichlet problem smooth on a time interval $[0,T)$ is studied. Smooth extendibility of the solution up to $t=T$ is proved, provided that “normilized local energies” of the solution are uniformly bounded on $[0,T)$. For the case where $[0,T)$ determines the maximal interval...

This work deals with a system of nonlinear parabolic equations arising in turbulence modelling. The unknowns are the N components of the velocity field u coupled with two scalar quantities θ and φ. The system presents nonlinear turbulent viscosity $A(\theta ,\varphi )$ and nonlinear source terms of the form ${\theta}^{2}{\left|\nabla u\right|}^{2}$ and ${\theta \varphi \left|\nabla u\right|}^{2}$ lying in L1. Some existence results are shown in this paper, including ${L}^{\infty}$-estimates and positivity for both θ and φ.

We classify the global behavior of weak solutions of the Keller-Segel system of degenerate and nondegenerate type. For the stronger degeneracy, the weak solution exists globally in time and has a uniform time decay under some extra conditions. If the degeneracy is weaker, the solution exhibits a finite time blow up if the data is nonnegative. The situation is very similar to the semilinear case. Some additional discussion is also presented.

The paper is devoted to mathematical modelling of erythropoiesis, production of red blood cells in the bone marrow. We discuss intra-cellular regulatory networks which determine self-renewal and differentiation of erythroid progenitors. In the case of excessive self-renewal, immature cells can fill the bone marrow resulting in the development of leukemia. We introduce a parameter characterizing the strength of mutation. Depending on its value, leukemia will or will not develop. The simplest...

We prove ${L}^{2}$-maximal regularity of the linear non-autonomous evolutionary Cauchy problem$$\dot{u}\left(t\right)+A\left(t\right)u\left(t\right)=f\left(t\right)\phantom{\rule{1.0em}{0ex}}\text{for}\phantom{\rule{4.0pt}{0ex}}\text{a.e.}\phantom{\rule{4.0pt}{0ex}}t\in [0,T],\phantom{\rule{1.0em}{0ex}}u\left(0\right)={u}_{0},$$ where the operator $A\left(t\right)$ arises from a time depending sesquilinear form $\U0001d51e(t,\xb7,\xb7)$ on a Hilbert space $H$ with constant domain $V.$ We prove the maximal regularity in $H$ when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance...