In a ball $\Omega \subset \mathbb {R}^{n}$ with $n\ge 2$, the chemotaxis system

\[ \left\{ \begin{array}{@{}l} u_t = \nabla \cdot \big( D(u)\nabla u\big) + \nabla\cdot \big(\dfrac{u}{v} \nabla v\big), \\ 0=\Delta v - uv \end{array} \right. \]

$u$

$v$

$D\in C^{3}([0,\infty ))$

$0< D(\xi ) \le {K_D} (\xi +1)^{-\alpha }$

$\xi >0$

${K_D}>0$

$\alpha >0$

$\Omega$

We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system:

\begin{eqnarray*} \left\{ \begin{array}{ll} u_t = u_{xx} - \chi \big(\frac{u}{v} \partial_x v \big)_x - uv +B_1(x,t), \qquad & x\in \Omega, \ t>0, \\[1mm] v_t = v_{xx} +uv - v + B_2(x,t), \qquad & x\in \Omega, \ t>0, \end{array} \right. \end{eqnarray*}

$\chi=2$

In bounded intervals $\Omega\subset\mathbb{R}$ and with prescribed suitably regular non-negative functions $B_1$ and $B_2$ , we first prove the existence of global classical solutions for any choice of $\chi>0$ and all reasonably regular non-negative initial data.

We next address the issue of determining the qualitative behaviour of solutions under appropriate assumptions on the asymptotic properties of $B_1$ and $B_2$ . Indeed, for arbitrary $\chi>0$ , we obtain boundedness of the solutions given strict positivity of the average of $B_2$ over the domain; moreover, it is seen that imposing a mild decay assumption on $B_1$ implies that u must decay to zero in the long-term limit. Our final result, valid for all $\chi\in\left(0,\frac{\sqrt{6\sqrt{3}+9}}{2}\right),$ which contains the relevant value $\chi=2$ , states that under the above decay assumption on $B_1$ , if furthermore $B_2$ appropriately stabilises to a non-trivial function $B_{2,\infty}$ , then (u,v) approaches the limit $(0,v_\infty)$ , where $v_\infty$ denotes the solution of

\begin{eqnarray*} \left\{ \begin{array}{l} -\partial_{xx}v_\infty + v_\infty = B_{2,\infty}, \qquad x\in \Omega, \\[1mm] \partial_x v_{\infty}=0, \qquad x\in\partial\Omega. \end{array} \right. \end{eqnarray*}

$\chi$

$\chi$

This study considers a model for oncolytic virotherapy, as given by the reaction–diffusion–taxis system

\[\begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla (u\nabla v)-\rho uz, \\ v_t = - (u+w)v, \\ w_t = D_w \Delta w - w + uz, \\ z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. \end{eqnarray*}\]

Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if β < 1, whereas whenever β > 1 and $({1}/{|\Omega |})\int _\Omega u(\cdot ,0) > 1/(\beta -1)$, infinite-time blow-up occurs at least in the particular case when ρ = 0.

In order to provide an appropriate complement to this, the current study reveals that for any ρ ⩾ 0 and arbitrary β > 0, at each prescribed level γ ∈ (0, 1/(β − 1)+) one can identify an L∞-neighbourhood of the homogeneous distribution (u, v, w, z) ≡ (γ, 0, 0, 0) within which all initial data lead to globally bounded solutions that stabilize towards the constant equilibrium (u∞, 0, 0, 0) with some u∞ > 0.

We propose and study a class of parabolic-ordinary differential equation models involving chemotaxis and haptotaxis of a species following signals indirectly produced by another, non-motile one. The setting is motivated by cancer invasion mediated by interactions with the tumour microenvironment, but has much wider applicability, being able to comprise descriptions of biologically quite different problems. As a main mathematical feature constituting a core difference to both classical Keller–Segel chemotaxis systems and Chaplain–Lolas type chemotaxis–haptotaxis systems, the considered model accounts for certain types of indirect signal production mechanisms. The main results assert unique global classical solvability under suitably mild assumptions on the system parameter functions in associated spatially two-dimensional initial-boundary value problems. In particular, this rigorously confirms that at least in two-dimensional settings, the considered indirectness in signal production induces a significant blow-up suppressing tendency also in taxis systems substantially more general than some particular examples for which corresponding effects have recently been observed.

In a planar smoothly bounded domain $\Omega$ , we consider the model for oncolytic virotherapy given by

$$\left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v) - uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = d_w \Delta w - w + uz, \\[1mm] z_t = d_z \Delta z - z - uz + \beta w, \end{array} \right.$$

$ D_w $

$ D_z $

$\beta$

$\beta \lt 1$

$M \gt 0$

$\beta \gt 0$

$$u\ge M \qquad \mbox{in } \Omega\times (0,\infty).$$

$\beta \gt 1$

$$\liminf_{t\to\infty} \inf_{x\in\Omega} u(x,t)\le \frac{1}{\beta-1}.$$

$\beta$

$\beta = 1$

Ab initio total energy calculations based on the local density approximation (LDA) and using a conjugate-gradient solver for the Kohn-Sham equations have been performed for cordierite, brucite, (Mg(OH)2) and diaspore (AlOOH). The calculated fractional coordinates of all structures are in good agreement with experimental diffraction data. The angle of the non-linear hydrogen bond in diaspore is reproduced well. The Raman active OH stretching frequency in brucite has been calculated using the frozen phonon approach and the calculated stretching frequency is in very good agreement with the observed value. The energetically most favourable calculated orientation of the proton-proton vector of an H2O molecule incorporated in the structural channels of cordierite agrees with findings deduced from spectroscopic data, and the calculated energy of hydration is in reasonable agreement with calorimetric data. It is therefore concluded that ab initio total energy calculations can confidently be used to predict properties of hydrogen bonded structures, which is difficult with conventional parameterized static lattice energy minimization calculations. An extension to the model is necessary to improve the agreement of the predicted to the observed lattice parameters for small structures.

The paper studies large time behaviour of solutions to the Keller–Segel system with quadratic degradation in a liquid environment, as given by

under Neumann boundary conditions in a bounded domain Ω ⊂ ℝn, where n ≥ 1 is arbitrary. It is shown that whenever U : Ω × (0,∞) → ℝn is a bounded and sufficiently regular solenoidal vector field any non-trivial global bounded solution of (⋆) approaches the trivial equilibrium at a rate that, with respect to the norm in either of the spaces L1(Ω) and L∞(Ω), can be controlled from above and below by appropriate multiples of 1/(t + 1). This underlines that, even up to this quantitative level of accuracy, the large time behaviour in (⋆) is essentially independent not only of the particular fluid flow, but also of any effect originating from chemotactic cross-diffusion. The latter is in contrast to the corresponding Cauchy problem, for which known results show that in the n = 2 case the presence of chemotaxis can significantly enhance biomixing by reducing the respective spatial L1 norms of solutions.

Background: Neurosurgical residents face a unique combination of challenges, including long duty hours, technically challenging cases, and uncertain employment prospects. We sought to assess the demographics, interests, career goals, self-rated happiness, and overall well-being of Canadian neurosurgery residents. Methods: A cross-sectional survey was developed and sent through the Canadian Neurosurgery Research Collaborative to every resident enrolled in a Canadian neurosurgery program as of April 1, 2016. Results: We analyzed 76 completed surveys of 146 eligible residents (52% response rate). The median age was 29 years, with 76% of respondents being males. The most popular subspecialties of interest for fellowship were spine, oncology, and open vascular neurosurgery. The most frequent self-reported number of worked hours per week was the 80- to 89-hour range. The majority of respondents reported a high level of happiness as well as stress. Sense of accomplishment and fatigue were reported as average to high and overall quality of life was low for 19%, average for 49%, and high for 32%. Satisfaction with work-life balance was average for 44% of respondents and was the only tested domain in which significant dissatisfaction was identified (18%). Overall, respondents were highly satisfied with their choice of specialty, choice of program, surgical exposure, and work environment; however, intimidation was reported in 36% of respondents and depression by 17%. Conclusions: Despite a challenging residency and high workload, the majority of Canadian neurosurgery residents are happy and satisfied with their choice of specialty and program. However, work-life balance, employability, resident intimidation, and depression were identified as areas of active concern.

Background Currently, the literature lacks reliable data regarding operative case volumes at Canadian neurosurgery residency programs. Our objective was to provide a snapshot of the operative landscape in Canadian neurosurgical training using the trainee-led Canadian Neurosurgery Research Collaborative. Methods: Anonymized administrative operative data were gathered from each neurosurgery residency program from January 1, 2014, to December 31, 2014. Procedures were broadly classified into cranial, spine, peripheral nerve, and miscellaneous procedures. A number of prespecified subspecialty procedures were recorded. We defined the resident case index as the ratio of the total number of operations to the total number of neurosurgery residents in that program. Resident number included both Canadian medical and international medical graduates, and included residents on the neurosurgery service, off-service, or on leave for research or other personal reasons. Results: Overall, there was an average of 1845 operative cases per neurosurgery residency program. The mean numbers of cranial, spine, peripheral nerve, and miscellaneous procedures were 725, 466, 48, and 193, respectively. The nationwide mean resident case indices for cranial, spine, peripheral nerve, and total procedures were 90, 58, 5, and 196, respectively. There was some variation in the resident case indices for specific subspecialty procedures, with some training programs not performing carotid endarterectomy or endoscopic transsphenoidal procedures. Conclusions: This study presents the breadth of neurosurgical training within Canadian neurosurgery residency programs. These results may help inform the implementation of neurosurgery training as the Royal College of Physicians and Surgeons residency training transitions to a competence-by-design curriculum.

We study the asymptotic behaviour of solutions of the fast diffusion equation near extinction. For a class of initial data, the asymptotic behaviour is described by a singular Barenblatt profile. We complete previous results on rates of convergence to the singular Barenblatt profile by describing a new phenomenon concerning the difference between the rates in time and space.

This paper deals with the coupled chemotaxis-haptotaxis model of cancer invasion given by

where χ, ξ and μ are positive parameters and Ω ⊂ ℝn (n ≥ 1) is a bounded domain with smooth boundary. Under zero-flux boundary conditions, it is shown that, for any μ > χ and any sufficiently smooth initial data (u0, w0) satisfying u0 ≥ 0 and w0 > 0, the associated initial–boundary-value problem possesses a unique global smooth solution that is uniformly bounded. Moreover, we analyse the stability and attractivity properties of the non-trivial homogeneous equilibrium (u, v, w) ≡ (1,1, 0) and establish a quantitative result relating the domain of attraction of this steady state to the size of μ. In particular, this will imply that whenever u0 > 0 and 0 < w0 < 1 in there exists a positive constant μ* depending only on χ, ξ, Ω, u0 and w0 such that for any μ < μ* the above global solution (u, v, w) approaches the spatially uniform state (1, 1, 0) as time goes to infinity.

We consider an anisotropic diffusion equation of the form ut = ∇∇(D(x)u) in two dimensions, which arises in various applications, including the modelling of wolf movement along seismic lines and the invasive spread of certain brain tumours along white matter neural fibre tracts. We consider a degenerate case, where the diffusion tensor D(x) has a zero-eigenvalue for certain values of x. Based on a regularisation procedure and various pointwise and integral a priori estimates, we establish the global existence of very weak solutions to the degenerate limit problem. Moreover, we show that in the large time limit these solutions approach profiles that exhibit a Dirac-type mass concentration phenomenon on the boundary of the region in which diffusion is degenerate, which is quite surprising for a linear diffusion equation. The results are illustrated by numerical examples.

We investigated the effects of sample preparation and of the exposure to an electron beam on particles in cryo-electron tomographs. Various virus particles with icosahedral symmetry were examined, allowing a comparison of symmetrically related components that should be identical in structure but might be affected differently by these imaging artifacts. Comparison of tomographic reconstructions with previously determined structures established by an independent method showed that neither freezing nor electron beam exposure produced a significant amount of shrinkage along the z axis (thickness). However, we observed damage to regions of the particles located close to the surface of the vitreous ice.

We consider large-time behaviour of global solutions of the Cauchy problem for a parabolic equation with a supercritical nonlinearity. It is known that the solution is global and unbounded if the initial value is bounded by a singular steady state and decays slowly. In this paper we show that the grow-up of solutions can be arbitrarily slow if the initial value is chosen appropriately.