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I examine some analytical properties of a nonlinear reaction-diffusion system that has been used to model the propagation of a wildfire. I establish global-in-time existence and uniqueness of bounded mild solutions to the Cauchy problem for this system given bounded initial data. In particular, this shows that the model does not allow for thermal blow-up. If the initial temperature and fuel density also satisfy certain integrability conditions, the spatial 
$L^2$-norms of these global solutions are uniformly bounded in time. Additionally, I use a bootstrap argument to show that small initial temperatures give rise to solutions that decay to zero as time goes to infinity, proving the existence of initial states that do not develop into traveling combustion waves.
In this article, we investigate a free boundary problem for the Lotka–Volterra model consisting of an invasive species with density u and a native species with density v in one dimension. We assume that v undergoes diffusion and growth in 
$[0,+\infty )$, and u invades into the environment with spreading front 
$x=h(t)$ satisfying free boundary condition 
$h'(t)=-u_x(t,h(t))-\alpha $ for some decay rate 
$\alpha>0$, this is caused by the bad environment at the boundary. When u is an inferior competitor, 
$u(t,x)$ and 
$h(t)$ tend to 0 within a finite time, while another specie 
$v(t,x)$ tends to a stationary 
$\Lambda (x)$ defined on the half-line. When u is a superior competitor, we have a trichotomy result: spreading of u and vanishing of v (i.e., as 
$t \to +\infty $, 
$h(t)$ goes to 
$+\infty $ and 
$(u,v)\to (\Lambda ,0)$); the transition case (i.e., as 
$t \to +\infty $, 
$(u,v)\to (w_\alpha , \eta _\alpha )$, 
$h(t)$ tends to a finite point); vanishing of u and spreading of v (i.e., 
$u(t,x)$ and 
$h(t)$ tends to 0 within a finite time, 
$v(t,x)$ converges to 
$\Lambda (x)$). Additionally, we show that this trichotomy result depends on the initial data 
$u(0,x)$.
In the second part of this series of papers, we address the same evolution problem that was considered in part 1 (see [16]), namely 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, 
$\phi (y) \equiv H\left (\frac {1}{4}-y^2\right )$, except that now we modify this to an associated initial-boundary value problem on the finite spatial interval 
$[0,a]$ rather than the whole real line. Boundary conditions are required at the end points of the interval, and we address the situations when these are of either Dirichlet or Neumann type. This model is a natural extension of the classical Fisher-KPP model, with the introduction of the simplest possible nonlocal effect into the saturation term. Nonlocal reaction-diffusion models arise naturally in a variety of (frequently biological or ecological) contexts, and as such it is of fundamental interest to examine their properties in detail, and to compare and contrast these with the well known properties of the classical Fisher-KPP model.
Some qualitative properties of radially symmetric solutions to the non-homogeneous heat equation with critical density and weighted sourceare obtained, in the range of exponents 
\begin{align*} |x|^{-2}\partial _tu=\Delta u+|x|^{\sigma }u^p, \quad (x,t)\in {\mathbb {R}}^N\times (0,T), \end{align*}
$p\gt 1$, 
$\sigma \ge -2$. More precisely, we establish conditions fulfilled by the initial data in order for the solutions to either blow-up in finite time or decay to zero as 
$t\to \infty$ and, in the latter case, we also deduce decay rates and large time behavior. In the limiting case 
$\sigma =-2$, we prove the existence of non-trivial, non-negative solutions, in stark contrast to the homogeneous case. A transformation to a generalized Fisher–KPP equation is derived and employed in order to deduce these properties.
In this paper, we prove the global exstence of weak solutions for a porous medium dynamics of m species moving between two domains separated by a zero-thickness membrane. On this membrane, Kedem–Katchalsky conditions are considered, and the study is characterized by natural structural conditions applied to the nonlinear reactive terms. The global existence is established under the assumption that these reactive terms are bounded in 
$L^1$. This problem has already been analyzed in the linear diffusion case by Ciavolella and Perthame in Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540). The present work constitutes an extension for nonlinear diffusion, particularly of the porous medium type, in the form 
$\partial _t v_i - \Delta v_i^{r_i} = R_i$, for an exponent 
$r_i < 2$. The case 
$r_i \geq 2$ remains an open problem. This paper is an adaptation of the ideas from Ciavolella and Perthame (2021, Journal of Evolution Equations 21, 1513–1540), with new strategies to overcome the appearance of nonlinearity and degeneracy in the diffusion term.
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.
