In this paper, we consider the dynamical behaviour of a reaction–diffusion model for a population residing in a one-dimensional habit, with emphasis on the effects of boundary conditions and protection zone. We assume that the population is subjected to a strong Allee effect in its natural domain but obeys a monostable nonlinear growth in the protection zone $[L_1,\, L_2]$ with two constants satisfying $0\leq L_1< L_2$, and the general Robin condition is imposed on $x=0$ (i.e. $u(t,\,0)=bu_x(t,\,0)$ with $b\geq 0$). We show the existence of two critical values $0< L_*\leq L^*$, and prove that a vanishing–transition–spreading trichotomy result holds when the length of protection zone is smaller than $L_*$; a transition–spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$; only spreading happens when the length of protection zone is larger than $L^*$. Based on the properties of $L_*$, we obtain the precise strategies for an optimal protection zone: if $b$ is large (i.e. $b\geq 1/\sqrt {-g'(0)}$), the protection zone should start from somewhere near $0$; while if $b$ is small (i.e. $b< 1/\sqrt {-g'(0)}$), then the protection zone should start from somewhere away from $0$, and as far away from $0$ as possible.

We propose amixed spectral method for heat transfer in unbounded domains, using generalised Hermite functions and Legendre polynomials. Some basic results on the mixed generalised Hermite-Legendre orthogonal approximation are established, which plays important roles in spectral methods for various problems defined on unbounded domains. As an example, the mixed generalised Hermite-Legendre spectral scheme is constructed for anisotropic heat transfer. Its convergence is proven, and some numerical results demonstrate the spectral accuracy of this approach.

The stochastic volatility jump diffusion model with jumps in both return and volatility leads to a two-dimensional partial integro-differential equation (PIDE). We exploit a fast exponential time integration scheme to solve this PIDE. After spatial discretization and temporal integration, the solution of the PIDE can be formulated as the action of an exponential of a block Toeplitz matrix on a vector. The shift-invert Arnoldi method is employed to approximate this product. To reduce the computational cost, matrix splitting is combined with the multigrid method to deal with the shift-invert matrix-vector product in each inner iteration. Numerical results show that our proposed scheme is more robust and efficient than the existing high accurate implicit-explicit Euler-based extrapolation scheme.

In this paper, we consider the Cauchy problem

$$\left\{ \begin{align} & {{u}_{t}}=\Delta ({{u}^{m}}),\,\,\,\,\,x\in {{\mathbb{R}}^{N}},t>0,N\ge 3, \\ & u(x,0)={{u}_{0}}(x),\,\,\,\,\,x\in {{\mathbb{R}}^{N}}. \\ \end{align} \right.$$

We will prove that

(i) for ${{m}_{c}}\,<\,m,\,{{m}_{0}}\,<\,1,\,\left| u(x,\,t,m)-u(x,\,t,{{m}_{0}}) \right|\,\to \,0$ as $m\,\to \,{{m}_{0}}$ uniformly on every compact subset of ${{\mathbb{R}}^{N}}\,\times \,{{\mathbb{R}}^{+}}$, where ${{m}_{c}}\,=\,\frac{{{(N-2)}_{+}}}{N}$;

(ii) there is a ${{C}^{*}}$ that explicitly depends on $m$ such that

$${{\left\| u(\cdot ,\cdot ,m)-u(\cdot ,\cdot ,1) \right\|}_{{{L}^{2}}({{\mathbb{R}}^{N}}\times {{\mathbb{R}}^{+}})}}\le {{C}^{*}}\left| m-1 \right|.$$

The asymptotic stability of two types of invariant solutions under a curvature flow in the whole plane is studied. First, by extending the work of others, we prove that the stationary line with nonzero slope will attract the graphical curves which surround it. Then a similar property is obtained for the grim reaper.

In this paper, we study an upper bound of the fractal dimension of the exponential attractor for the chemotaxis–growth system in a two-dimensional domain. We apply the technique given by Eden, Foias, Nicolaenko and Temam. Our results show that the bound is estimated by polynomial order with respect to the chemotactic coefficient in the equation similar to our preceding papers.

This paper is devoted to the long-time behavior of solutions to the Cauchy problem of the porous medium equation $u_t=\Delta(u^m)-u^p$ in $\mathbb{R}^n\times (0,\infty)$ with $(1-2/n)_{+}<m<1$ and the critical exponent $p=m+2/n$. For the strictly positive initial data $u(x,0)=O(1+|x|)^{-k}$ with $n+mn(2-n+nm)/(2[2-m+mn(1-m)])\leq k<2/(1-m)$, we prove that the solution of the above Cauchy problem converges to a fundamental solution of $u_t=\Delta(u^m)$ with an additional logarithmic anomalous decay exponent in time as $t\to \infty$.

an approximation theorem in inhomogeneous orlicz–sobolev spaces is proved which allows a second-order parabolic equation in orlicz spaces to be solved. a trace result is also given which shows that the solutions are continuous with respect to time.

Consider the parabolic equation \[ \begin{array}{@{}l@{\qquad}l@{}} u_t=\Delta u+|u|^p& \text{in}\ R^d\times(0,T), u(x,0)=\phi(x)& \text{in}\ R^d, \end{array} \] up to a maximal time $T=T_\infty$, where $p>1$. Let $p^{\ast}=1+2/d$. It is a classical result that if $p\le p^{\ast}$, then there exist no non-negative, global solutions to the above equation for any choice of $\phi\geq0$; that is, necessarily $T_\infty<\infty$ and the solution blows up in some sense. On the other hand, if $p>p^{\ast}$, then there do exist non-negative, global solutions for appropriate choices of $\phi$. A recent paper by Zhang seems to be the first to consider the existence of global solutions for the above equation with sign-changing initial data. He proved that if $p\le p^{\ast}$ and $\int_{R^d}\phi(x)dx>0$, then the solution to the above equation is not global. In this paper it is shown that this result continues to hold if $\int_{R^d}\phi(x)dx=0$, and then it is shown that for each $p{>}1$ and each $l\in(-\infty,0)$, there exists a $\phi$ satisfying $\int_{R^d}\phi(x)dx=l$ and such that the corresponding solution to the above equation is global, and there exists a $\phi$ satisfying $\int_{R^d}\phi(x)dx=l$ and such that the corresponding solution is non-global.

We obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time $t$ and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones.