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In this paper, we prove the existence and regularity of pullback attractors for non-autonomous nonclassical diffusion equations with nonlocal diffusion when the nonlinear term satisfies critical exponential growth and the external force term 
$h \in L_{l o c}^{2}(\mathbb {R} ; H^{-1}(\Omega )).$ Under some appropriate assumptions, we establish the existence and uniqueness of the weak solution in the time-dependent space 
$\mathcal {H}_{t}(\Omega )$ and the existence and regularity of the pullback attractors.
We consider the no-flux initial-boundary value problem for the cross-diffusive evolution system:which was introduced by Short et al. in [40] with 
\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*}
$\chi=2$
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 We conclude with some numerical simulations exploring possible effects that may arise when considering large values 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*}
$\chi$
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.
In this study, we investigate the intial value problem (IVP) for a time-fractional fourth-order equation with nonlinear source terms. More specifically, we consider the time-fractional biharmonic with exponential nonlinearity and the time-fractional Cahn–Hilliard equation. By using the Fourier transform concept, the generalized formula for the mild solution as well as the smoothing effects of resolvent operators are proved. For the IVP associated with the first one, by using the Orlicz space with the function 
$\Xi (z)={\textrm {e}}^{|z|^{p}}-1$ and some embeddings between it and the usual Lebesgue spaces, we prove that the solution is a global-in-time solution or it shall blow up in a finite time if the initial value is regular. In the case of singular initial data, the local-in-time/global-in-time existence and uniqueness are derived. Also, the regularity of the mild solution is investigated. For the IVP associated with the second one, some modifications to the generalized formula are made to deal with the nonlinear term. We also establish some important estimates for the derivatives of resolvent operators, they are the basis for using the Picard sequence to prove the local-in-time existence of the solution.
This paper is concerned with spreading phenomena of the classical two-species Lotka-Volterra reaction-diffusion system in the weak competition case. More precisely, some new sufficient conditions on the linear or nonlinear speed selection of the minimal wave speed of travelling wave fronts, which connect one half-positive equilibrium and one positive equilibrium, have been given via constructing types of super-sub solutions. Moreover, these conditions for the linear or nonlinear determinacy are quite different from that of the minimal wave speeds of travelling wave fronts connecting other equilibria of Lotka-Volterra competition model. In addition, based on the weighted energy method, we give the global exponential stability of such solutions with large speed 
$c$. Specially, when the competition rate exerted on one species converges to zero, then for any 
$c>c_0$, where 
$c_0$ is the critical speed, the travelling wave front with the speed 
$c$ is globally exponentially stable.
We will give a new proof of the existence of hypercylinder expander of the inverse mean curvature flow which is a radially symmetric homothetic soliton of the inverse mean curvature flow in 
$\mathbb {R}^{n}\times \mathbb {R}$
, 
$n\ge 2$
, of the form 
$(r,y(r))$
or 
$(r(y),y)$
, where 
$r=|x|$
, 
$x\in \mathbb {R}^{n}$
, is the radially symmetric coordinate and 
$y\in \mathbb {R}$
. More precisely, for any 
$\lambda>\frac {1}{n-1}$
and 
$\mu>0$
, we will give a new proof of the existence of a unique even solution 
$r(y)$
of the equation 
$\frac {r^{\prime \prime }(y)}{1+r^{\prime }(y)^{2}}=\frac {n-1}{r(y)}-\frac {1+r^{\prime }(y)^{2}}{\lambda (r(y)-yr^{\prime }(y))}$
in 
$\mathbb {R}$
which satisfies 
$r(0)=\mu $
, 
$r^{\prime }(0)=0$
and 
$r(y)>yr^{\prime }(y)>0$
for any 
$y\in \mathbb {R}$
. We will prove that 
$\lim _{y\to \infty }r(y)=\infty $
and 
$a_{1}:=\lim _{y\to \infty }r^{\prime }(y)$
exists with 
$0\le a_{1}<\infty $
. We will also give a new proof of the existence of a constant 
$y_{1}>0$
such that 
$r^{\prime \prime }(y_{1})=0$
, 
$r^{\prime \prime }(y)>0$
for any 
$0<y<y_{1}$
, and 
$r^{\prime \prime }(y)<0$
for any 
$y>y_{1}$
.
