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We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,where 
\begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*}
$\phi *u$ is a spatial convolution with the top hat kernel, 
$\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$. After observing that the problem is globally well-posed, we demonstrate that positive, spatially periodic solutions bifurcate from the spatially uniform steady state solution 
$u=1$ as the diffusivity, 
$D$, decreases through 
$\Delta _1 \approx 0.00297$ (the exact value is determined in Section 3). We explicitly construct these spatially periodic solutions as uniformly valid asymptotic approximations for 
$D \ll 1$, over one wavelength, via the method of matched asymptotic expansions. These consist, at leading order, of regularly spaced, compactly supported regions with width of 
$O(1)$ where 
$u=O(1)$, separated by regions where 
$u$ is exponentially small at leading order as 
$D \to 0^+$. From numerical solutions, we find that for 
$D \geq \Delta _1$, permanent form travelling waves, with minimum wavespeed, 
$2 \sqrt{D}$, are generated, whilst for 
$0 \lt D \lt \Delta _1$, the wavefronts generated separate the regions where 
$u=0$ from a region where a steady periodic solution is created via a distinct periodic shedding mechanism acting immediately to the rear of the advancing front, with this mechanism becoming more pronounced with decreasing 
$D$. The structure of these transitional travelling wave forms is examined in some detail.
In a smoothly bounded domain 
$\Omega \subset \mathbb{R}^n$, 
$n\ge 1$, this manuscript considers the homogeneous Neumann boundary problem for the chemotaxis systemwith parameter 
\begin{eqnarray*} \left \{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[5pt] v_t = \Delta v + u - \alpha uv, \end{array} \right . \end{eqnarray*}
$\alpha \gt 0$ and with coincident production and uptake of attractants, as recently emphasized by Dallaston et al. as relevant for the understanding of T-cell dynamics.
It is shown that there exists 
$\delta _\star =\delta _\star (n)\gt 0$ such that for any given 
$\alpha \ge \frac{1}{\delta _\star }$ and for any suitably regular initial data satisfying 
$v(\cdot, 0)\le \delta _\star$, this problem admits a unique classical solution that stabilizes to the constant equilibrium 
$(\frac{1}{|\Omega |}\int _\Omega u(\cdot, 0), \, \frac{1}{\alpha })$ in the large time limit.
In this paper, we consider the following PDE-ODE system modelling cancer invasion with slow diffusion and ECM remodelling,\[ \begin{cases} u_t=\Delta u^m-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla\omega)+\mu u(1-u-\omega), \\ v_t=\Delta v+u-v, \\ \omega_t={-}v\omega+\eta \omega(1-u-\omega). \end{cases} \]For the special case $\eta =0$
, fruitful results have been achieved since Tao and Winkler's work in 2011. However, there is no any progress for the general case $\eta >0$
in the past ten years. In this paper, we analysed some commonly used research methods when $\eta =0$
, and found that these methods are completely unsuitable for situations where $\eta >0$
. By introducing some new forms of functionals, we reconstruct the relationship between the haptotactic term and the nonlinear diffusion term, and ultimately prove the global existence of weak solutions. This result improves and perfects a series of works previously presented in the literature.
This paper investigates the separation property in binary phase-segregation processes modelled by Cahn-Hilliard type equations with constant mobility, singular entropy densities and different particle interactions. Under general assumptions on the entropy potential, we prove the strict separation property in both two and three-space dimensions. Namely, in 2D, we notably extend the minimal assumptions on the potential adopted so far in the literature, by only requiring a mild growth condition of its first derivative near the singular points 
$\pm 1$, without any pointwise additional assumption on its second derivative. For all cases, we provide a compact proof using De Giorgi’s iterations. In 3D, we also extend the validity of the asymptotic strict separation property to the case of fractional Cahn-Hilliard equation, as well as show the validity of the separation when the initial datum is close to an ‘energy minimizer’. Our framework offers insights into statistical factors like particle interactions, entropy choices and correlations governing separation, with broad applicability.
This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant\[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \]in a bounded domain $\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$
with smooth boundary $\partial \Omega$
. It is shown that if $m>\max \{1,\,\frac {3N-2}{2N+2}\}$
, for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium $(\bar {u}_0,\,0)$
in an appropriate sense as $t\rightarrow \infty$
, where $\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$
. This result not only partly extends the previous global boundedness result in Fan and Jin (J. Math. Phys. 58 (2017), 011503) and Wang and Xiang (Z. Angew. Math. Phys. 66 (2015), 3159–3179) to $m>\frac {3N-2}{2N}$
in the case $N\geq 3$
, but also partly improves the global existence result in Zheng and Wang (Discrete Contin. Dyn. Syst. Ser. B 22 (2017), 669–686) to $m>\frac {3N}{2N+2}$
when $N\geq 2$
.
Existence of specific eternal solutions in exponential self-similar form to the following quasilinear diffusion equation with strong absorption\[ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, \]posed for $(t,\,x)\in (0,\,\infty )\times \mathbb {R}^N$
, with $m>1$
, $q\in (0,\,1)$
and $\sigma =\sigma _c:=2(1-q)/ (m-1)$
is proved. Looking for radially symmetric solutions of the form\[ u(t,x)={\rm e}^{-\alpha t}f(|x|\,{\rm e}^{\beta t}), \quad \alpha=\frac{2}{m-1}\beta, \]we show that there exists a unique exponent $\beta ^*\in (0,\,\infty )$
for which there exists a one-parameter family $(u_A)_{A>0}$
of solutions with compactly supported and non-increasing profiles $(f_A)_{A>0}$
satisfying $f_A(0)=A$
and $f_A'(0)=0$
. An important feature of these solutions is that they are bounded and do not vanish in finite time, a phenomenon which is known to take place for all non-negative bounded solutions when $\sigma \in (0,\,\sigma _c)$
.
For $s\in [\tfrac {1}{2},\, 1)$
, let $u$
solve $(\partial _t - \Delta )^s u = Vu$
in $\mathbb {R}^{n} \times [-T,\, 0]$
for some $T>0$
where $||V||_{ C^2(\mathbb {R}^n \times [-T, 0])} < \infty$
. We show that if for some $0<\mathfrak {K} < T$
and $\epsilon >0$
\[ {\unicode{x2A0D}}-_{[-\mathfrak{K},\, 0]} u^2(x, t) {\rm d}t \leq Ce^{-|x|^{2+\epsilon}}\ \forall x \in \mathbb{R}^n, \]then $u \equiv 0$
in $\mathbb {R}^{n} \times [-T,\, 0]$
.
We investigate an inverse boundary value problem of determination of a nonlinear law for reaction-diffusion processes, which are modeled by general form semilinear parabolic equations. We do not assume that any solutions to these equations are known a priori, in which case the problem has a well-known gauge symmetry. We determine, under additional assumptions, the semilinear term up to this symmetry in a time-dependent anisotropic case modeled on Riemannian manifolds, and for partial data measurements on 
${\mathbb R}^n$.
Moreover, we present cases where it is possible to exploit the nonlinear interaction to break the gauge symmetry. This leads to full determination results of the nonlinear term. As an application, we show that it is possible to give a full resolution to classes of inverse source problems of determining a source term and nonlinear terms simultaneously. This is in strict contrast to inverse source problems for corresponding linear equations, which always have the gauge symmetry. We also consider a Carleman estimate with boundary terms based on intrinsic properties of parabolic equations.
