We propose a method that combines Isogeometric Analysis (IGA) with the interior penalty discontinuous Galerkin (IPDG) method for solving the Allen-Cahn equation, arising from phase transition in materials science, on three-dimensional (3D) surfaces consisting of multiple patches. DG ideology is adopted at patch level, i.e., we employ the standard IGA within each patch, and employ the IPDG method across the patch interfaces. IGA is very suitable for solving Partial Differential Equations (PDEs) on (3D) surfaces and the IPDG method is used to glue the multiple patches together to get the right solution. Our method takes advantage of both IGA and the IPDG method, which allows us to design a superior semi-discrete (in time) IPDG scheme. First and most importantly, the time-consuming mesh generation process in traditional Finite Element Analysis (FEA) is no longer necessary and refinements, including h-refinement and p-refinement which both maintain the original geometry, can be easily performed at any level. Moreover, the flexibility of the IPDG method makes our method very easy to handle cases with non-conforming patches and different degrees across the patch interfaces. Additionally, the geometrical error is eliminated (for all conic sections) or significantly reduced at the beginning due to the geometric flexibility of IGA basis functions, especially the use of multiple patches. Finally, this method can be easily formulated and implemented. We present our semi-discrete IPDG scheme after generally describe the problem, and then briefly introduce the time marching method employed in this paper. Theoretical analysis is carried out to show that our method satisfies a discrete energy law, and achieves the optimal convergence rate with respect to the L2 norm. Furthermore, we propose an elliptic projection operator on (3D) surfaces and prove an approximation error estimate which are vital for us to obtain the error estimate in the L2 norm. Numerical tests are given to validate the theory and gauge the good performance of our method.

We prove the existence of weak solutions for the strongly nonlinear parabolic problem

in the anisotropic Sobolev space , where the data f are assumed to be in the dual, and the nonlinear term g(x, t, s) has growth and sign conditions on s.

We extend the Atiyah, Patodi, and Singer index theorem for first-order differential operators from the context of manifolds with cylindrical ends to manifolds with periodic ends. This theorem provides a natural complement to Taubes’ Fredholm theory for general end-periodic operators. Our index theorem is expressed in terms of a new periodic eta-invariant that equals the Atiyah–Patodi–Singer eta-invariant in the cylindrical setting. We apply this periodic eta-invariant to the study of moduli spaces of Riemannian metrics of positive scalar curvature.

We present a new method for investigating the Lp-type pullback attractors (2 ≤ p < ∞) of a semilinear heat equation on a time-varying domain under quite general assumptions on the nonlinear and forcing terms. The existing approach does not appear applicable here as it is impossible to show the existence of a pullback absorbing set in Lp space when p is large. A new asymptotic decomposition scheme for a non-autonomous pullback attractor has been introduced. The abstract results and preliminary lemmas are also of independent interest and applicable to other systems.

We consider the quasi-periodic solutions bifurcated from a degenerate homoclinic solution. Assume that the unperturbed system has a homoclinic solution and a hyperbolic fixed point. The bifurcation function for the existence of a quasi-periodic solution of the perturbed system is obtained by functional analysis methods. The zeros of the bifurcation function correspond to the existence of the quasi-periodic solution at the non-zero parameter values. Some solvable conditions of the bifurcation equations are investigated. Two examples are given to illustrate the results.

This paper is concerned with the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts for time-periodic reaction–diffusion equations with bistable nonlinearity in ℝm with m ≥ 2. It should be mentioned that the existence and stability of two-dimensional time-periodic V-shaped travelling fronts and three-dimensional time-periodic pyramidal travelling fronts have been studied previously. In this paper we consider two cases: the first is that the wave speed of a one-dimensional travelling front is positive and the second is that the one-dimensional wave speed is zero. For both cases we establish the existence, non-existence and qualitative properties of cylindrically symmetric travelling fronts. In particular, for the first case we furthermore show the asymptotic behaviours of level sets of the cylindrically symmetric travelling fronts.

We study the asymptotic behaviour as t → T–, near a finite blow-up time T > 0, of decreasing-in-x solutions to the following semilinear heat equation with a non-local term:

with Neumann boundary conditions and strictly decreasing initial function u0(x) with zero mass. We prove sharp estimates for u(x, t) as t → T–, revealing a non-uniform global blow-up:

uniformly on any compact set [δ, 1], δ ∈ (0, 1).

We study non-autonomous parabolic equations with critical exponents in a scale of Banach spaces Eσ, σ ∈ [0,1 + μ). We consider a suitable E1+ε-solution and describe continuation properties of the solution. This concerns both a situation when the solution can be continued as an E1+ε-solution and a situation when the E1+ε-norm of the solution blows up, in which case a piecewise E1+ε-solution is constructed.

Given a domain ${\rm\Omega}$ of a complete Riemannian manifold ${\mathcal{M}}$, define ${\mathcal{A}}$ to be the Laplacian with Neumann boundary condition on ${\rm\Omega}$. We prove that, under appropriate conditions, the corresponding heat kernel satisfies the Gaussian upper bound

$$\begin{eqnarray}h(t,x,y)\leq \frac{C}{[V_{{\rm\Omega}}(x,\sqrt{t})V_{{\rm\Omega}}(y,\sqrt{t})]^{1/2}}\biggl(1+\frac{d^{2}(x,y)}{4t}\biggr)^{{\it\delta}}e^{-d^{2}(x,y)/4t}\quad \text{for}~t>0,~x,y\in {\rm\Omega}.\end{eqnarray}$$

$d$

${\mathcal{M}}$

$V_{{\rm\Omega}}(x,r)$

$B(x,r)\cap {\rm\Omega}$

$B(x,r)$

$x$

$r$

${\it\delta}$

${\rm\Omega}$

$e^{-t{\mathcal{A}}}$

$L^{p}({\rm\Omega})$

$p\in [1,\infty )$

In this paper we consider the problem of existence of mild solutions to semilinear fractional heat equations with non-local initial conditions. We provide sufficient conditions for existence and regularity of such solutions.

This paper is devoted to the study of the local existence, uniqueness, regularity, and continuous dependence of solutions to a logistic equation with memory in the Bessel potential spaces.

We consider the first boundary value problem for a second-order parabolic equation with variable coefficients in the domain $K\times \mathbb{R}^{n-m}$, where $K$ is an $m$-dimensional cone. The main results of the paper are pointwise estimates of the Green’s function.

Let $u$ be a supertemperature on an open set $E$, and let $v$ be a related temperature on an open subset $D$ of $E$. For example, $v$ could be the greatest thermic minorant of $u$ on $D$, if it exists. Putting $w=u$ on $E\setminus D$ and $w=v$ on $D$, we investigate whether $w$, or its lower semicontinuous smoothing, is a supertemperature on $E$. We also give a representation of the greatest thermic minorant on $E$, if it exists, in terms of PWB solutions on an expanding sequence of open subsets of $E$ with union $E$.  In addition, in the case of a nonnegative supertemperature, we prove inequalities that relate reductions to Dirichlet solutions. We also prove that the value of any reduction at a given time depends only on earlier times.

Let $(X,d,{\it\mu})$ be a metric measure space satisfying the doubling, reverse doubling and noncollapsing conditions. Let $\mathscr{L}$ be a nonnegative self-adjoint operator on $L^{2}(X,d{\it\mu})$ satisfying a pointwise Gaussian upper bound estimate and Hölder continuity for its heat kernel. In this paper, we introduce the Hardy spaces $H_{\mathscr{L}}^{p}(X)$, $0<p\leq 1$, associated to $\mathscr{L}$ in terms of grand maximal functions and show that these spaces are equivalently characterised by radial and nontangential maximal functions.

We prove continuity in generalized parabolic Morrey spaces of sublinear operators generated by the parabolic Calderón—Zygmund operator and by the commutator of this operator with bounded mean oscillation (BMO) functions. As a consequence, we obtain a global -regularity result for the Cauchy—Dirichlet problem for linear uniformly parabolic equations with vanishing mean oscillation (VMO) coefficients.

In this paper we discuss an exponential integrator scheme, based on spatial discretization and time discretization, for a class of stochastic partial differential equations. We show that the scheme has a unique stationary distribution whenever the step size is sufficiently small, and that the weak limit of the stationary distribution of the scheme as the step size tends to 0 is in fact the stationary distribution of the corresponding stochastic partial differential equations.

We show that the null limit hypothesis, in the definition of a barrier, can be relaxed for normal boundary points that satisfy a mild additional condition. We also give a simple necessary and sufficient condition for the regularity of semi-singular boundary points.

We propose and analyse an alternate approach to a priori error estimates for the semidiscrete Galerkin approximation to a time-dependent parabolic integro-differential equation with nonsmooth initial data. The method is based on energy arguments combined with repeated use of time integration, but without using parabolic-type duality techniques. An optimal $L^2$-error estimate is derived for the semidiscrete approximation when the initial data is in $L^2$. A superconvergence result is obtained and then used to prove a maximum norm estimate for parabolic integro-differential equations defined on a two-dimensional bounded domain.