On the global existence and qualitative behaviour of one-dimensional solutions to a model for urban crime

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*} which was introduced by Short et al. in [40] with $\chi=2$ to describe the dynamics of urban crime. 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*} We conclude with some numerical simulations exploring possible effects that may arise when considering large values of $\chi$ not covered by our qualitative analysis. We observe that when $\chi$ increases, solutions may grow substantially on short time intervals, whereas only on large timescales diffusion will dominate and enforce equilibration.

Ground state solutions to a class of critical Schrödinger problem

We consider the following critical nonlocal Schrödinger problem with general nonlinearities − ε 2 a + ε b ∫ R 3 | ∇ u | 2 Δ u + V ( x ) u = f ( u ) + u 5 , x ∈ R 3 , u ∈ H 1 ( R 3 ) , $$\begin{array}{} \displaystyle \left\{\begin{array}{} &-\left(\varepsilon^{2}a+\varepsilon b\displaystyle\int\limits_{\mathbb{R}^{3}}|\nabla u|^{2}\right){\it\Delta} u+V(x)u=f(u)+u^{5}, &x \in \mathbb{R}^{3},\\ &u\in H^{1}(\mathbb{R}^{3}), \end{array}\right. \end{array}$$ (SKε ) and study the existence of semiclassical ground state solutions of Nehari-Pohožaev type to (SK ε ), where f(u) may behave like |u| q–2 u for q ∈ (2, 4] which is seldom studied. With some decay assumption on V, we establish an existence result which improves some exiting works which only handle q ∈ (4, 6). With some monotonicity condition on V, we also get a ground state solution v̄ ε and analysis its concentrating behaviour around global minimum x ε of V as ε → 0. Our results extend some related works.

Symmetry of solutions for a fractional p-Laplacian equation of Choquard type

Let [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] be a positive solution of the equation [Formula: see text] We prove that if [Formula: see text] satisfies some decay assumption at infinity, then [Formula: see text] must be radially symmetric and monotone decreasing about some point in [Formula: see text]. Instead of using equivalent fractional systems, we exploit a generalized direct method of moving planes for fractional [Formula: see text]-Laplacian equations with nonlocal nonlinearities. This new approach enables us to cover the full range [Formula: see text] in our results.

Existence of a Positive Solution to a Nonlinear Scalar Field Equation with Zero Mass at Infinity

We establish the existence of a positive solution to the problem-\Delta u+V(x)u=f(u),\quad u\in D^{1,2}(\mathbb{R}^{N}),for {N\geq 3}, when the nonlinearity f is subcritical at infinity and supercritical near the origin, and the potential V vanishes at infinity. Our result includes situations in which the problem does not have a ground state. Then, under a suitable decay assumption on the potential, we show that the problem has a positive bound state.

Analysis of positive solutions for classes of quasilinear singular problems on exterior domains

We consider the problem\left\{\begin{aligned} \displaystyle{-}\Delta_{p}u&\displaystyle=K(x)\frac{f(u% )}{u^{\delta}}&&\displaystyle\text{in }\Omega^{e},\\ \displaystyle u(x)&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\\ \displaystyle u(x)&\displaystyle\to 0&&\displaystyle\text{as }|x|\to\infty,% \end{aligned}\right.where {\Omega\subset\mathbb{R}^{N}} ({N>2}) is a simply connected bounded domain containing the origin with {C^{2}} boundary {\partial\Omega}, {\Omega^{e}:=\mathbb{R}^{N}\setminus\overline{\Omega}} is the exterior domain, {1<p<N} and {0\leq\delta<1}. In particular, under an appropriate decay assumption on the weight function K at infinity and a growth restriction on the nonlinearity f, we establish the existence of a positive weak solution {u\in C^{1}(\overline{\Omega^{e}})} with {u=0} pointwise on {\partial\Omega}. Further, under an additional assumption on f, we conclude that our solution is unique. Consequently, when Ω is a ball in {\mathbb{R}^{N}}, for certain classes of {K(x)=K(|x|)}, we observe that our solution must also be radial.

Uniform Bounds of Aliasing and Truncated Errors in Sampling Series of Functions from Anisotropic Besov Class

Errors appear when the Shannon sampling series is applied to approximate a signal in real life. This is because a signal may not be bandlimited, the sampling series may have to be truncated, and the sampled values may not be exact and may have to be quantized. In this paper, we truncate the multidimensional Shannon sampling series via localized sampling and obtain the uniform bounds of aliasing and truncation errors for functions from anisotropic Besov class without any decay assumption. The bounds are optimal up to a logarithmic factor. Moreover, we derive the corresponding results for the case that the sampled values are given by a linear functional and its integer translations. Finally we give some applications.

Effects of Repeated VI Reinforcement and Extinction upon Operant Behavior

Four pigeons trained on a 1-min. VI schedule were reinforced, extinguished, and reconditioned under each of two stimulus conditions for 20 daily sessions. The average rate in extinction decreased significantly as a function of the number of extinctions. A concurrent increase in rate occurred during VI reinforcement periods. Extinction occurred more rapidly as it was repeated. Assuming an exponential decay in rate of responding during extinction, the effects of repetition upon the parameters of fitted curves was examined. The decay constant (ratio of deceleration to rate) increased linearly as a function of the number of extinctions, while the log initial rate decreased linearly. These changes in extinction parameters were in the same direction as previously obtained in an experiment involving another species and procedure, but which did not permit determination of the form of relations between extinction parameters and the number of extinctions. The empirical grounds for the exponential decay assumption are discussed in connection with the effects of the principal independent variables influencing the course of operant extinction.